It is possible not only for a pinned piece to block a king's motions, but also for a pinned piece to actively give check or even checkmate. For example:
[FEN "r/6bb/8/n/1k/2N/P3QB/K6r w - - 0 0"]
1. Be1 Bxc3+ 2. Qb2+ Nb3++
White's bishop blocks the check, pinning itself. Black's bishop pins itself when taking the knight, but simultaneously checks White. Note that White's bishop is pinned, so it can't take Black's bishop. White's Queen blocks check from the bishop, pinning itself, but also checking Black. Because Black's bishop is pinned, Black cannot take White's queen (and although White's queen is pinned, it could take Black's bishop next turn and Black would not be able to re-take). Black's knight, however, can deliver checkmate while blocking check and pinning itself in line with White's queen. Note that White's pawn cannot take Black's knight because the knight's last move pinned it.