If one does something on Shabbos, say, that is patur aval issur (like an act of destruction) what is the punishment? I know that Beis Din (court) does nothing, but are there any sources for what the punishment from heaven would be, if any?
Beis din could inflict makus mardus. That could in fact be more than the 39 lashes he gets for a biblical prohibition when the punishment isn't death (such as eating treifa or shaving with a razor). And according to the Gemara he gets מיתה בידי שמים (death by God's hand, not beis din's) if he violates rabbinic prohibitions to show, lhachus (deliberately), that it's only rabbinic and that he doesn't care about rabbinically set boundaries.
Rambam, Shabas 1:3, says:
Someone who does so on purpose, we hit him with a smiting for rebellion (makas mardus).
That is, bes din does.