Both natural selection and genetic drift play a part in changing our DNA over time. Are there trustworthy estimates how much of the differences over the whole genome between us and chimpanzees are due to genetic drift and how much is due to natural selection?
The easiest way to track whether a particular protein-coding gene has diverged between two species because of drift or actively maintained selection is to look at the ratio of synonymous to nonsynonymous mutations within that gene. A synonymous mutation is one that doesn't change the identity of the amino acid within that protein; a nonsynonymous mutation is one that does. Synonymous mutations are neutral by definition; their presence gives us an idea of how much time has diverged and how frequently mutations in that protein coding region change over time.
A dN/dS (Nonsynonymous mutation to Synonymous mutation) ratio of 1 suggests that the primary driver of change between species is purely neutral change. That usually means that it doesn't matter what is going on with that protein and is a little unusual--I would primarily expect to see something like that in a gene that is on its way to becoming a pseudogene.
A ratio of less than one means that synonymous mutations are much more likely to persist in that region than nonsynonymous ones. This is usually a sign that purifying selection is happening--for whatever reason, it's important to the function of both species that those proteins stay relatively similar.
A ratio of more than 1 means that there is diverging or directional selection happening, and that that protein type is changing between the two species because selection is actively modifying one or both proteins.
Wildman et al. (2003) investigated 97 genes in humans and chimpanzees and show evidence that 30 of them (~31%) have evolved under positive selection. This is not per say an estimate of what fraction of substitutions are adaptive but it gives an idea.
Such estimates can be quite hard to get. I don't know any such estimates between human and primates but I know one on Drosophila.
Eyre-Walker and Keightley (2009) that about 40% of all substitutions between Drosophila melanogaster and D. simulans are adaptive.
Population sizes being larger in the Drosophila lineage than in the Hominidae lineage, I would predict that the fraction of substitutions that are adaptative would be significantly lower between humans and chimpanzees (prediction which agrees with Wildman et al. (2003)).
So, to the question
How much of the genetic differences between humans and chimpanzees are due to genetic drift?, I would answer: More than 60%! But remain cautious realizing this is a vague estimate.