<p>I assume SLA printing only works on Earth and upright.</p>
<p>But would fused-filament printer (e.g. Prusa Mk3) work in zero gravity? What about upside down or sideways? If not, could it be modified to work in other orientations? Have there been any demonstrations of it?</p>

Yes!
3D Printing upside would only potentially have an issue with the first layer if you're using an extremely large gap on the first layer, however in normal circumstances there's enough pressure that the filament is squished into the bed, if you've got one yourself you can put it on it's side, the question when it comes to 3D Printing
isn't
upside down, but not the correct way, as it's presuming that gravity pushing down is the important thing, while in every other orientation it also works.
In theory SLA printing would work just fine, as long as you're able to seal the build plate and vat together so no resin leaks out.