If a Pokémon is already locked into a move via a held Choice item, if the effect of its held item is negated (by Klutz, Embargo, or Magic Room), it is free to select any move it wants, but moves other than the move it is locked into fail ("But it failed" message with no details on why).
Note that you cannot become locked into a move via a held Choice item while under the effect of Klutz/Embargo/Magic Room, but if (for example) you use Role Play while holding a Choice Band on a Pokémon with Klutz, you will become locked into Role Play (although the Pokémon is free to select any move, everything but Role Play will fail).
If a Pokémon is locked into a move via Choice item, but loses that item while its effect is negated (e.g. via Magic Room), it will remain locked into a move and other moves will continue to fail. If the Pokémon's held item regains its effect (e.g. Magic Room wears off), as long as it has an item regardless of what the held item is, the Pokémon will be unable to select moves other than the one it is locked into. Attempting to select other moves will display "The <HELD ITEM> only allows the use of <MOVE>!" (in my test, I had it say "The Rare Candy only allows the use of Spore!"). If it has no held item in this case, moves other than the locked move will fail.
Original source is Smogon Sun and Moon research thread
, but I have supplemented my post with my own research.
This is a bit tricky to describe, so it would be really helpful if someone produced a video showing this off.