As a citizen of Warsaw, I never carry cash around (and most young people don't), since card payments are accepted almost everywhere (as long as you use Visa or Mastercard), even in small stores, especially in the central parts of Warsaw (which is what you're most likely going to see as a tourist). I regularly visit restaurants, and not being able to pay by card happened to me like once in the last 5 years, and that was because the terminal was broken. I know of one bakery chain that still requires cash, but otherwise this was never a problem. On the other hand, there are situations where it's a huge inconvenience to use cash (for example ticket vending machines present in pretty much every bus accept only card payments, so if you only had cash, you'd have to buy the ticket from the driver).
Minimum amounts are prohibited, though I've seen this a few times (usually 10 PLN in stores, doesn't happen in restaurants) - this is becoming much rarer than it used to (last time I saw this was two years ago). According to my bank, if I ever see someone requiring a minimum amount, I should ask for terminal ID (which they are obliged to disclose) and report the situation - so, the risk here is pretty low.
As for museums, they also almost always accept credit card payments, especially the major ones.
If you want to be on the safe side, get 50 to 100 PLN in cash - you're most likely never going to need it, though.
Update: As of November 2018, buying tickets in a bus by cash is no longer possible, and the bakery chain which was cash only now accepts cards as well.