If you're running a 1/8 cog then you'll have to run a 1/8 chain on the 3/32 chain ring which will have some play.

More flex in a 3/32 chainring.

If the whole drive train is 3/32 you will run a wimpy chain and be a wimp.. jk

Well there is less material in the 3/32 chainring so it will wear faster.

But these aren't really big disadvantages.