Gypsum drywall is effectively an insulator at ordinary household voltages, as long as it is dry. Its bulk electrical resistance is in the range of dry wood. We are talking about megohms to tens of megohms, or current leakages in the microamp range.
In the shop, we work on power supplies that have live 120 volt component legs sticking through the underside of the circuit board. We often use little 12 inch square panels of scrap drywall to protect the board from accidental contact with any bits of fine wire or solder that could be on the workbench itself, and as a fire-proof barrier in case there is something seriously enough wrong with the unit under test that it spits sparks and fire when powered up.
So, unless you are running something with truly lethal voltages and currents, e.g. 480 volt 3 phase, or your drywall is positioned where someone can spill a drink onto the material amid energized wiring, it ought to be fine.
Other pretty good insulators include Plexiglas (acrylic plastic), Lexan (polycarbonate plastic), and HardiPanel (cement board). Of those the plastics have the highest total dielectric strength.