Ok I guess I will start with some background info. I bought the Nes off ebay for $27. It arrived in a little tiny box with no peanuts and was extremely hot when I took it out of the box. I had it hooked up via AV cords so I no it is not the RF switch. I hope that it is just the 72 pin connector and not the circuit board.
Does the Power light blink on and off?
Does the sound work?
If it's a toaster NES (i.e. first model), try sliding the game sideways while it's pushed down in the NES and hit reset. You can also try dipping a Q-Tip in some rubbing alcohol (70% or more concentrated, higher numbers are better) and swabbing the contacts in the game cart to clean them. Then dry them with the dry end of the Q-Tip.
a trick i found back in the days was to wiggle the cart (with the power off, and set down into "play mode") by gripping the sides with your pointer fingers and wiggling the cartridge while in the slot from left to right. it worked for me 99% of the time!