Since I moved recently, I've been unable to play any game full-screen: my monitor will flicker on and off after a few minutes, like someones hitting the power button repeatedly. The only thing that's running hot is the chipset on my motherboard, which had a fan die on it. I removed the fan, but now the heatsink is hot enough to burn to the touch. Are there any cheap solutions to this?
Firstly, are you saying the display is faulty, or the motherboard?
If you moved recently, the jarring of the move could have shook some dust loose and its laying on components inside the case, the display, or both. Consider taking the case apart on both the computer and the monitor, and carefully use compressed air to clean both. CAUTION, however. Be very careful inside a monitor case, since high voltages may still be present. Also, don't use compressed air to "spin up" fans since that will generate voltage back into the power system for the fans, possible damaging something. When cleaning computer fans with compressed air, use a plastic rod or something to hold the fan blades still.
Any running of a heatsinked chip without the original heatsink fan is foolish and you must stop using that IMMEDIATELY before the chip dies. There's no way around it; you're going to have to replace the fan.
In rare instances, like with my Dell Optiplex machines, the CPU and power supply fans will go lube-dry and grind slower, until they halt. About once a year I take 'em out and apply actual lithium hi-temp grease to the fine shafts of these fans. That takes some doing, but the fans will last for years whereas they otherwise would just look dead. Possibly this kind of thing applies to the heatsink fan you removed.