Just wanted to mention that I was suffering from the same problem… hot (very) and noisy (very) NUC. Has been running pretty much continuously for a few years, but this problem has been growing over the last six months or so.
I tried the tried-and-true compressed canned air. And it helped. But not enough. Noise and heat came back in a few months. So I went ahead and disassembled it. The instructions in this thread helped, but were honestly not required. Anyone that builds PCs could figure it out pretty quickly. Otherwise you might want to get help from someone that does.
Anyway, once I got it opened, and the motherboard removed, on the underside of the motherboard was the fan, which was easily removed by removing two small screws. I do want to mention that these screws were installed in the factory, and were pretty snug. I think using a Phillips head screwdriver that was not the EXACT right would have lead to stripped screws. Luckily I had such a screwdriver. And when the fan was removed (be careful of the power wire attaching it the the MB), I was amazed to see how much dust was lodged in the spot where the fan intake meets the long, internal, rectangular air intake vent. I’m talking about a LOT of thick dust.
30 seconds with a tight nozzled vacuum cleaner hose and the dust was history. 60 seconds later the entire thing was reassembled, and running super cool. And silent. And I’m expecting that it will remain that way for a few years once again.
Best of luck to anyone with a similar problem.