You may not realize, but you actually have two questions here.
Question number 1 is this:
Can birds raise younglings that are not their own?
The answer is yes. We call this Brood parasitism, and it occurs in several species of birds.
There is both intraspecific (between members of the same species) and interspecific (between members of different species) brood parasitism. I believe you want to know about interspecific brood parasitism.
By far the most famous examples are the cuckoos (Family Cuculidae). Some species of cuckoos are obligate parasites, meaning that they only reproduce laying eggs in other birds' nests.
It's very funny (for us, probably not for the host) the fact that sometimes the young parasite is way bigger than the host mum:
A Eurasian reed warbler raising a common cuckoo. Source: Wikipedia
Then, we come to your question number 2, which is different:
If a scientist were to move a chickadee or pigeon egg from one nest to another, would the new bird (second mother) in the new nest incubate or raise the youngling?
The answer here depends on several factors, mainly the species involved.
For species that suffer broad parasitism, there is egg-rejection if the parasite egg is somehow different. If you look at the adaptations to parasitism in the Wikipedia link I shared, you're gonna see that "among specialist avian brood parasites, mimetic eggs are a nearly universal adaptation".
However, species that don't suffer broad parasitism are more prone to accept egg transfer (see @bshane comment here).