Would using litz wire for an inductor change the inductance compared to regular wire?
Litz wire inductance should be the same for un-stranded and stranded-but-uninsulated wire.

It should also be noted that Litz wire reduces current proximity effect. But.......

EDIT

I sat down and thought about this a bit more and it occurred to me that skin effect (the phenomenon that increases a wire's resistive losses with frequency) must also slightly change the inductance and this effect could be different for Litz wire compared to other types of wire so I'm posting this picture I got from this site here and maybe someone can give a more definitive answer than mine: -



EDIT 2

For a given total conductor cross sectional area, the internal eddy currents in solid wire that drive current to the "skin" must be smaller in Litz wire. Those internal eddy currents don't represent loss but, represent a reduction in inductance therefore, I would say, that at those frequencies where Litz wire is more effective than solid wire, the inductance of Litz wire should be slightly higher.