Do formic acid and phosphorus pentoxide react at room temperature?
As noted, the dehydrating action of H2SO4 on HCOOH liberates CO gas. To quote a source:

The sulfuric acid removes the elements of water (i.e., H2O) from the formic or oxalic acid and absorbs the water produced.

As P2O5 is at least as dehydrating as Sulfuric acid, expect the same per the reaction:
$$\ce{P2O5 + 3 HCOOH -&gt; 3 CO (g) + 2 H3PO4}$$