Should I increase contributions to HSA or 401(k)?

You will be allowed to exhaust the funds in the HSA, but the same rules apply (qualified medical expenses, etc). Some HSA's will allow you to invest once the balance gets high enough, but it may be a limited choice of funds, so you may be better off just putting it in your 401k.

Some HSA plans do offer investment options, once you reach a certain minimum balance. Mine does, but they're limited and some have higher than average fees. Also, with Optum Bank (our HSA provider), trying to find a way to speak to a living, breathing human is all but impossible. I would sock away enough in the HSA to cover the deductible or max out of pocket for a couple of years and then devote additional contributions to the 401k. If you have, or expect to have, heavy medical expenses in the short term, obviously YMMV.