The .22-250 is based on a .308 Winchester case,
No, it is not.
It is based on the .250 Savage case. Which was introduced in 1915. The .22-250 was created as a wildcat round in the 1930s, and was quite popular, eventually becoming standardized and adopted by Remington about 1965. It shares the same .473" head size as many other rounds, developed originally by Mauser for their 8mm and 7mm rounds and used by the .30-06 and .308 Win family of cartridges, and even the .45ACP since Mauser introduced it in 1888.
The .378" head size of the .223 Rem comes from the .222 Rem case, introduced by Remington in 1950.
Is the .22-250 a barrel burner? It can be, if pushed to the max. Lots of high performance rounds are, some worse than others.
I think an AR 10 in .22-250 would be an interesting project, if you like trying to solve problems with the AR type rifles. Mechanically, the two big issues I see are feeding from the magazine (previously mentioned) and even more important, properly timing the cycle of the action.
At a minimum, I'd think an adjustable gas system would be needed, and then there is the issue of whether or not existing stock parts such as action spring and buffer (made for larger calibers) would be suitable. Theory says they should be, but reality sometimes doesn't do what theory says it should.
Good Luck if you go that route, and please, do let us know what you encounter.