If Alice can win, Bob should attempt to force a draw. If Bob can win, Alice should attempt to force a draw. So long as either player can force a draw, the game must end in a draw. Either player can indeed force a draw by making every line unable to be completed. You can use the "L" pieces along the edge of the board to make a 1x2 or 2x1 "nook" of empty space than cannot ever be filled in, making it impossible to complete that line (or two lines). There is no way to block the edge of the board to prevent this from happening, since if you put a piece there, it becomes the new board edge against which a "nook" can be created. Either player is able to create an unfillable hole on every line, making it impossible to complete any line, and forcing the game to end in a draw.
EDIT: I think this proof is a little incomplete, since unlike Tetris, the grid doesn't have to fill from the bottom. If you put an L piece like a 7 in the bottom left corner, for example, the opposition could attempt to complete Line 3, but there's not a very obvious way to orphan any Line 3 squares in one move if Lines 1 and 2 are mostly empty.