Why does my ceiling lights require some other outlet in the room to be turned on and off again before it works?
This is NOT normal.

If I had to guess, I would agree with Steve; there is a connection somewhere in this circuit that is just barely loose, and is requiring an electrical arc to complete the circuit. The light bulb just isn't enough, but turn on that power-hungry heating element and the circuit is now drawing enough for the arcing to be the "easiest path". Arcing across a gap produces heat which helps to swell the metal parts in the connection, drawing them closer together so this doesn't occur once the circuit has "warmed up".

If I were a betting man, I would put a nickel on your home having aluminum electrical wiring. Aluminum is more susceptible to this expansion and contraction while under load, which is why copper is used in virtually all new construction. Aluminum isn't dangerous, per se, you just need to know what it will do over time and plan for it; terminals should be screwed down very tightly, a little slack should be given in the box and along wire runs, and of course Al-rated switches, outlets and receptacles should be used.

Before calling an electrician there are some things you can try yourself. With the power off at the breaker, open up the switch box and outlets, and clamp down all the wiring terminals on both switch and outlet. If the outlet or switch seems very old, beat-up, worn, rusted, or generally untrustworthy, replace it. If you do have aluminum wire (should be real easy to tell), it will require Al-rated switches and outlets; they can handle the higher heat, the expansion and contraction of aluminum as it heats, and are made of metals that won't react galvanically with aluminum. DO NOT use switches rated for "copper or copper-clad wire only"; the switch or outlet will say exactly that on the back of the switch body, and will have a circle and line drawn through the letters "Al".