The solution is actually easier than you might think, but it requires a bit of skill in disassembling the time machine extreme, a couple of simple tools (something to pry the device base open, a #8 torx bit screwdriver that is at least 6" long, and a bottle of lubricant such as bicycle chain lubricant or something similar).
Use a disassembly guide to get the device torn down. Once you get the internals out, you'll find the fan on one side of the device. It is covered, but you can open up the cover by popping the little clips on the sides of the fan. Once you've exposed the blades of the fan, give it a little spin. You'll notice it doesn't keep spinning. The bearing is jammed. Now here's the trick - grab the fan from the sides, and lift it STRAIGHT UP. It'll slight right off. Now put a drop of oil in the hole the fan blade shaft goes into, and shove the fan into the hole a few times to lubricate it. Give it a spin. It'll spin for awhile. You fixed it! Reassemble and there will be no more overheating!
A few quick additions: When I mention lifting the fan "straight up" I mean STRAIGHT UP from the base. When you open the fan case, you'll see the top of the fan and the blades around the sides. Grip on the sides and lift it straight up from the plastic base it's attached from (don't try to pry it or shove it to the sides). You'll have a tiny bit of resistance (because the shaft is jammed and because there's a bit of magnetism holding the fan in place). This is almost universally the reason this overheating error occurs - the fan isn't spinning or is barely spinning. One tiny drop of oil solves the entire problem. It only takes a LITTLE oil, so be sparing in how you do it (you might even want to lube the shaft instead of the hole). The hardest part of the whole job is disassembling the device, but all-in-all this is a 15 minute procedure and will fix the problem.