As long as you have good draw the stove should run under negative pressure.
Does your stove backpuff or smell like smoke when its in operation? Is it hard to light?
If you have chimney problems then yes, it could definatly affect your skimmer. If you dont believe me, take a twig, light it up blow it out so that it smokes and put it next to your skimmer. The foam will colapse.
Once again, if you stove draws well it can actually improve tank conditions, it will help increase PH with the fresh air that the wood stove brings in, and the wood stove will help keep your tank warm during a power outage.