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The Tree of Life is still up for debate. Most of this debate seems to be due to a lack of genomic information, but that deficiency is decreasing rapidly with advances in technology and sequencing power.
Hypothetically, if we knew the genome of every species, would we resolve all phylogenetic debates? If not, what would still be up for debate?
Horizontal gene transfer
Don't expect to have a tree! Horizontal gene transfer happens and therefore we would end up with a network, not a tree.
Gene trees
Different DNA sequences have different evolutionary histories. See, in particular, the question of incomplete lineage sorting. This means that one may compute a tree for a given DNA sequence that must disagree with the tree of another sequence. So, don't expect a perfect species tree where all sequences will agree on.
Note that the two points "Horizontal gene transfer" and "Gene trees" are highly related. So much so that they could fit together.
Not all individuals have been discovered
If by saying of every organism, you meant "of every organism that we encounter", then we still have the issue of organisms that we have never encountered. If you meant "absolutely every individual that exist", then this is a non-issue. It also matters as to whether you mean every individual of every species or just one individual per species.
Also, it is slightly unclear whether, in your hypothesis, we can sequence all dead individuals! Extinct lineages would still not be resolved if we cannot sequence them.
Massive mutational events
It is possible that very large mutational event could prevent us to see the exact relationship between individuals. But that would only interfere on an extremely fine scale phylogeny (intra-species level).
Methodology
If we really sample all the individuals, the detail of the methodology may not matter much. It will still need to be decided whether we want to consider AATC closer to AA.C or to AGTC (where . indicates a deletion). Again, this will likely only matter on a very fine scale (intra-species level).
Computational power
Of course, if we sequence every nucleotide that exists on earth, we would never have the computational power to even store the data. Not talking about actually processing it. We would not either have the computational power to process the sequencing (nor the necessary amount of products to allow for the sequencing to happen).
If you meant to fully sequence only one individual per species (and then again the arbitrary nature of the concept of species will complicate things up), then we might have the computational power (see @KonradRudolf's comment)
The concept of species can often be misleading. You might want to read the post How could humans have interbred with Neanderthals if we're a different species? for a discussion on the concept of species.
A quick (but not quite glib) answer (having worked in the field): Given identical sequences of DNA from well-known species with likely close relationships - in other words, comparing beetles with other beetles, rather than snakes or bananas - respected researchers almost never arrive at the same phylogenetic trees, especially with very large sequences of DNA, even if they are working with exactly the same DNA samples.
There are several reasons.
Researchers have differing opinions about the weight to give the likelihood of different possible mutations that could have resulted in the differences between the DNA sequences. This automatically results in different phylogenetic trees. This leads to:
There are innumerable paths that could lead to the differences in DNA sequences. This means that there is no way of knowing the actual path that in fact led to any given difference, much less all of them that make up even two different species. This then means that there are innumerable different phylogenetic trees describing the possible ancestry of even two relatively similar species. Computerized tree generating and sorting algorithms are used to make sense of the huge numbers involved, but they cannot and do not list all of the possible trees, as that cannot be done. The list is infinite in size. Instead, they stop generating trees once they reach a pre-set limit (by the researcher, based on heuristics) of likelihood, then sort the ones they have. It's not as random as it sounds, but there is a degree of guesswork in any phylogenetic tree. This could only be mitigated by knowing the DNA sequences of every single ancestor of every single organism so that we could see how the DNA changes at every base pair in each sequence in each generation. It is a little like asking for time travel, but the math is harder.
There is fundamental disagreement, from species to species, about which changes in DNA make for different species and which merely represent variation in a single species. In reality, there is approximately one mutation for every cell division (see this article on mutation rates in humans for a look at the scale of the problem1) across most known species, most of which are meaningless or are spontaneously removed via cell or (often, in the case of germ cells) organismal death. This is a real issue, discussed at length by systematic taxonomists.
Because the DNA of different species is actually different, it is not possible to achieve a one-to-one match of the DNA sequences between two species. Researchers make a best guess, based on what they think is most likely (and gets more difficult the more species are used).
Statistical methods (which are required by the infinite size of the data sets, even with (relatively) small amounts of DNA and numbers of species) only tell the likelihood of a particular tree. They cannot predict the actual phylogeny. This can only be known if all (or at least most) of the ancestors of each of the representative organisms' DNA can be examined for the mutations that resulted in speciation.
In short (because this answer really was not, despite what I originally said), DNA sequences of every extant species would not be enough to resolve phylogeny. DNA sequences of every extant individual organism would not be enough. Only knowing the DNA sequences of every organism that has ever lived would be enough to resolve phylogeny completely.
No, but it would help. First, though, let's talk about limitations. The limitation you speak of is not rapidly disappearing. The barcode initiative (attempting to sequence a small fragment of mt DNA from every organism on Earth) has been going strong for almost 20 years now and is not even close to covering every species on Earth. Many species have only been encountered once, and have not been seen since their description. Most species remain undescribed. Then there is the issue of computing power. To analyze the data for 10 million species (one ballpark estimate for the number of species on Earth), or even 1.8 million (current number) would not be possible. In addition, phylogeneticists are an argumentative bunch. It is likely there would be debates over methodology. There would also likely be errors in the handling of the data. There would be contamination issues, and labelling errors.
However, as you said, your question was hypothetical. A carefully, error-free analysis, assuming complete coverage and perfect models, and assuming we had the computing power to do it would resolve most of the debate. It would still not resolve species with reticulate evolution or hard polytomies (groups that simultaneously split into more than two lineages). For that, you would need more than one individual to represent a species, and even with say 10 per species, it would be difficult.
This question is a lot like asking "if we found every fossil there is, would we know everything about life in the past?" or "if we found every ancient manuscript that exists, would we know everything about history?". The answer for both of those is "no", because fossils and manuscripts are an inherently incomplete record. Not all organisms fossilize; not everything is written down; not all fossils or manuscripts survived until today, and those that did suffered various levels of decay; the longer it's been, the more they've decayed.
The same is true of the phylogenetic signal one can infer from genes. The reason we can infer ancestry from genes in the first place is that everyone gets their genes from their parents, and those genes are slightly modified each generation. This means that siblings have very similar genes, cousins have slightly less similar genes, and so on so forth until you have every living thing's family tree.
Except that several factors mess this up. Here are a few:
Everyone doesn't get their genes from their parents. Animals mostly do, but bacteria swap genes such that each gene can have its own family tree, independent of that of the bacteria it's in. This is extremely relevant to the origin of life in particular because it involved bacteria and archaea.
The "very similar, less similar, etc" ranking works if we have infinitely large genes, containing infinite information. If we don't, then at some point you get two individuals whose every base of DNA is different from one another, and from that point on they can't get more different, however distantly they're related. Also, with finite DNA you get mutations on top of mutations - you might have two distantly-related lineages that get the same mutation by chance, or an old mutation gets reversed, or a new mutation happens on top of an old one, looking like only one mutation. All of this can blur the "signal" of relatedness, and it's the reason phylogenetic trees are calculated using statistical methods and as much DNA information as is practical. We (most of us) have huge genomes, which is the reason we can make phylogenetic trees as good as the ones we do, but there are genomes small enough that they can have been completely shuffled since life began, and it's harder to build phylogenetic trees there. On the opposite end of the spectrum, organisms that are so closely related that there are few differences to work from can be hard to sort out too, simply because a single event like mutations overlapping or being reverted can have a bigger impact when it isn't drowned out by thousands of other differences.
The theory also relies on the mutations being random; every individual is different from their parent to a predictable extent because no factor makes some mutations more likely than others. This is the case for many parts of the genome, and that does make them pretty good for phylogenetics, but many other parts are selected for or against. Meaning some mutations are more likely to stick around because they're beneficial at some point, and others are more likely to get weeded out quickly because they're harmful. This is why, even in organisms with small genomes that could be entirely shuffled since life began, they aren't really entirely shuffled: the parts of the genome that are necessary for life stay the same. This can be good for examining deep phylogeny, because it means those sequences change a lot more slowly. It can also be bad, because now family relationships aren't the only reason they change: fitness also affects that and you can no longer be sure that two organisms have similar or different DNA because of their relatedness, or because they've adapted to the same or different functions. This depends on the DNA in question, whether the function can be done by many different DNA configurations or only one, but at the end of the day, it's another factor to muddle the relatedness signal. And while the slower rate of change in conserved sequences makes it possible to separate very deep phylogenies, by the same token it won't do so at a very high resolution, because there are fewer differences to work from.
So essentially, like fossils and manuscripts genomic information is an imperfect signal for building family trees, and it's a decaying one - the longer it's been, the more time there's been for the various factors that make it imperfect in the first place to work their magic. So to answer your question, having the genomic information for all species or organisms would help a lot with resolving the Tree of Life, and might even resolve all the current debates on it. But it might also not. Hard to know without actually resolving the debates. And even if it does, there would still probably be outstanding questions; the general outline of the picture would be clear but there would still be all kinds of details that weren't filled.
