I think you need to structure your recursion differently. I'm also not sure adding and removing pieces from different places of the list is safe; much as I'd rather avoid allocation in the recursion it might be safest to create a list copy, or scan the board so far for instances of the same piece to avoid re-use.
public Piece[][] solve(int r, int c, List<Piece> piecesLeft) {
// Note that this check is equivalent to
// 'have r and c gone past the last square on the board?'
// or 'are there no pieces left?'
if (isSolved())
return board;
// Try each remaining piece in this square
for (Piece p : piecesLeft) {
// in each rotation
for(int orientation = 0; orientation < 4; ++orientation) {
if (tryInsert(p, r, c, orientation)) {
// It fits: recurse to try the next square
// Create the new list of pieces left
List<Piece> piecesLeft2 = new ArrayList<Piece>(piecesLeft);
piecesLeft2.remove(p);
// (can stop here and return success if piecesLeft2 is empty)
// Find the next point
Point next = nextLoc(r, c);
// (could also stop here if this is past end of board)
// Recurse to try next square
Piece[][] solution = solve(next.x, next.y, piecesLeft2);
if (solution != null) {
// This sequence worked - success!
return solution;
}
}
}
}
// no solution with this piece
return null;
}