Does putting solder mask over a ground plane influences its heat exchange with the air?

The short answer is no it won’t make a significant difference. The long answer is that conduction will be slightly worse with solder mask but radiation will be better, assuming that the bare metal is in good condition - the emissivity of a shiny metal surface is generally low, around 0.1 while that of a coated surface is 0.9 or more. Thus bare metal may paradoxically be a worse heat sink at higher temperatures. This assumes that the bare metal doesn’t oxidise or corrode, unlikely if it’s copper but possible if it’s tinned or gold plated.

The solder mask coating is a very thin thermal and electrical insulator. However, "preferable" is a relative term. Exposed copper or solder plate will oxidize / corrode / whatever, and as above is uninsulated from accidental contact with a non-ground-potential conductor of some kind.

So, yes, there are several differences; but no, I've never seen a situation where the thermal difference is significant.