Another one of those things I should know but never really have the answer to.

If you put a player on non-revokable waivers, and he is claimed, does the new team pick up his contract or only the minimum.?

For some reason, I thought in real MLB that this was the risk teams took if claiming a player - that they'd have to pay his salary.  i.e. The Cubs put Soriano on waivers hoping someone will claim him and pay his salary - but no one claims him because they don't want his awful contract.

But I think in HD the original team is still on the hook for the contract minus the ML minimum.

Is this correct?  Or do teams who claim a player pick up the salary cost as well?

Entire contract. 

Yep, entire contract.

What may have been stuck in your mind is that if a player is "released" but picked up by another team, the 2nd team pays the league min ($327k) while the original team is on the hook for the rest of the contract.