What would happen if two Oricorio are allies in a doubles match and one of them uses swords dance? I know the other would copy it, but then would the one who first used it use it again? Would it go on forever?
No.
After the first Oricorio uses Swords Dance, the other copies it, then it ends.
Source
Dancer will not activate for a move copied by another Pokémon with Dancer, but it will activate on an instructed dance move.
No, it would not get re-copied