Not really. Unlike previous mac mini models, the grille on the underside is both the air intake and exhaust. If anything it ought to be better upside-down, since convection now helps the heated air rise out.
It would make it fill up with dust, even while idle. Also, and fluid spill would be much more likely to cause damage.
The normal orientation is fine for most people who want things to be as simple as possible (ie, most Mac users). There is very little reason to ever turn it off. If you still do that frequently for some reason, just leave it in a locatino where the button is still easy to access.
The case is also going to radiate heat and turning it upside down will make that less efficient. The base won't radiate heat in the same due to being plastic and due to it already being used to pass air through.
The case isn't thermally connected to the SoC in any direct way, so while I'm sure it does radiate some heat, I think it's pretty negligible. The PSU sits at the "top" (in regular orientation), and I don't think it runs much of a risk of overheating.
Keep the thing upside down.
Not joking.