What is the mechanism for the acid-catalyzed esterification of formic acid and 2-butanol?

Here is a likely mechanism. It's worth noting that you may have an elimination side product. Can you imagine how step 4 might branch off to form the side product?

(The intermediate there is a poor nucleophile by virtue of its bulk, but has two electron-donating groups that facilitate Lewis basic behavior. Surprisingly, that branched step also leads to formation of the major product!)

1. I assume you know how to write a proton transfer mechanistic step, from protonated 2-butanol to formic acid's carbonyl oxygen (which has the highest electron density on formic acid). If you don't, it is demonstrated in reverse on step 6.
2. The 2-butanol can nucleophilically attack the carbonyl carbon, since it's been activated by the acid catalysis and is more electrophilic.
3. Proton transfer pt 1.
4. Proton transfer pt 2.
5. Tetrahedral collapse. Either hydroxyl group could have been protonated; I just chose the one that makes the mechanism look more like the one you see in my textbook (Bruice).
6. Deprotonation of the carbonyl oxygen to yield the product.

It is hydronium that exists at the end of the mechanism, rather than protonated 2-butanol, because a protonated alcohol generally has a lower #"pKa"# than #-1.7#, the #"pKa"# of #"H"_3"O"^(+)# (the equilibrium lies on the side of the weaker acid).