Thought my space heater had died when I plugged it in and it wouldn't turn on, not even the "on" indicator light, so I went out and bought a new one. When the new one wouldn't work either, I tested the outlet and a lamp worked just fine. Carried the heater around the house and had no problem with the heater working in other outlets -- just not in the room where I want it!
First, before you open up the receptacle, I would use a power tester (you can get one at the hardware store) to check the voltage and grounding of both sockets. Your heater may have a safety circuit that requires correct wiring/grounding and keeps it from turning on if the socket is incorrectly grounded or wired.
A voltage tester/fault detector will tell you if each socket is wired correctly and if it is grounded correctly. It's not uncommon for some sockets to be wired backwards or for your ground wire to be loose, unconnected, or missing. test each socket, including those that "work" and those that don't and see if there is an obvious wiring fault.
If there is a wiring fault, THEN you can shut off power in the room, take the cover off of the receptacle, and inspect the sockets for damage, loose wires, or incorrect wiring.