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.