Xbox keeps showing an overheating warning then shutting off

It's dust until proven otherwise. Pull it out from wherever it lives, check the vents, and get a can of compressed air into the back and the intake. If it's in a cabinet, that's the problem, it needs air on all sides. Stand it the way the manual shows for your model, some of them vent badly lying down. If it's clean and in open air and still doing it, the fan is likely going or the thermal paste has dried, and that's an open-it-up job. Don't keep running it when it warns, that's how you cook the chip.