Proof: Any rectangle made of Tetris pieces must have an even number of squares (in fact, a multiple of 4) and hence the same number of black/white squares. Every Tetris piece except the T has the same number of black/white squares, hence the T cannot be used in any arrangement of a subset of Tetris pieces into a rectangle.
Every even x even rectangle has equal dark and light squares.
Every even x odd rectangle has equal dark and light squares.
Every odd x odd rectangle has one more odd then even, or one more even then odd. (1x1 square is obvious, any odd x odd rectangle can be reduced to 1x1 square by removing only odd x even rectangle.)
Since the T piece creates a odd-even difference of 2, any number of other tetris pieces (including repeats) that only contains a single T piece can never form a rectangle with no holes.