The Oxford Advanced Learner's Dictionary gives the following example sentence, along with its interpretation in brackets:
He never does more work than he can help (= he does as little as possible) .
Actually, the problem lies not so much with the meaning of "help" as with the structure.
Consider a structurally identical sentence:
He never does more work than he is given.
That means, for example, that if he is given 10 pieces of work, he will not do an eleventh. He only does what he is asked to do.
Back to the OP sentence. Suppose "help" means "avoid." Then the original sentence can be rephrased as below:
He never does more work than he can avoid (doing).
Bearing in mind that it means "he does as little as possible," we can see the puzzle is that the intended interpretation is actually the reverse of the above sentence:
He never does more work than he cannot avoid (doing).
I.e., he never does more work than he has to do; he only does as much as he has to do.