Which Oxygen atom in HCOOH (formic/methanoic acid) does Carbon donate its electrons to, to obtain a partial positive charge?
Electronic effects are cumulative. For example, the central carbon atom in an orthoester like HC(OEt)3 is surrounded by 3 oxygen atoms which add up.

In the case of formic acid (which is what you described in your question), yes the carbon atom is donating its electron density to BOTH oxygen atoms. However, the oxygen atom involved in the C=O bond will attract more electrons, because the double-bond is shorter.