Many single-time use keys encryption scheme (not Shamir's Shared Secret)
Since you already trust the server to securely delete the plaintext message after decrypting it for a client, you can trust it to delete the encrypted versions too.

So for instance the case of 3 keys K1 .. K3 and plaintext P, store K1(P), K1(K2(P)),K1(K2(K3(P))) since they all start with K1, that is the only key that can be used at first. after K1 is presented, apply it to everything in the series, so you end up with.
P, K2(P), K2(K3(P)). serve up P deleting it and store the rest as your new series for when K2 arrives. repeat the proceduce until the whole sequence and hence all the keys are used up.