Can I put an extension cord under a rug?

Yes, but only if it’s a flat cord specifically rated for under-carpet use. Round cords are a fire hazard under rugs because they trap heat. Flat cords dissipate heat better and are designed for this, but they carry less current (typically 13 amps vs 15 amps), so don’t run a space heater through one. Always check the package for “carpet” or “under rug” labeling.