Does Finkelstein's reaction proceed through an SN2 mechanism?
The Finkelstein reaction is a nucleophilic substitution reaction. As such, it generally has the choice of proceeding via the $\mathrm{S_N1}$ or the $\mathrm{S_N2}$ mechanism.

The reaction is typically conducted in acetone; a polar but not very protic solvent. The solvent is used because all the sodium halides except for sodium iodide are not very soluble in it and precipitate out. It does not show any real preference for either $\mathrm{S_N1}$ or $\mathrm{S_N2}$.

Therefore, we should study the nucleophile and leaving group. Iodide is both a good nucleophile and a good leaving group; the former is important as it allows us to assume a viable $\mathrm{S_N2}$ mechanism. The other halides that you are substituting with iodide are not as good leaving groups but still rather okay. This again does nothing really to tilt the scales towards either $\mathrm{S_N1}$ or $\mathrm{S_N2}$.

The final choice we have is that it depends on the substrate. Probably, if you are replacing a primary halide chances are that the reaction will proceed according to the $\mathrm{S_N2}$ mechanism while an $\mathrm{S_N1}$ mechanism should predominate for tertiary halides.