Certain problems/studies that are illegal in chess might be legal in chess960, but I can't give any specific examples.
While writing that post I didn't have time to hunt for examples, but they aren't too hard to discover. In Chess glossary for Freshman Seminar: Chess and Mathematics, I found
Legal position (n.): a position that can be reached from the initial array by game consisting entirely of legal moves, however bizarre. Conventionally every chess problem should have a legal position. Naturally then, an illegal position is a position that cannot be reached by a legal game. For instance, a position in which one side has more than 8 pawns, or has both White and Black Kings in check, is illegal (why?). So is any position with a White Bishop on a1 and White pawn on b2 (why?), such as the following mutual Zugzwang (q.v.), which Lewis Stiller discovered in the course of an exhaustive computer search: White Kg6, Bh1, Pg2; Black Kg4, Pg3.
The given position is shown in the following diagram.
[FEN "8/8/6K1/8/6k1/6p1/6P1/7B w - - 0 1"]
The tablebase used in Shredder Computer Chess - Endgame Database accepts the position as valid and gives WTM 'Draw' and BTM 'Lose in 20'.