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.
travel.stackexchange.com/users/67266/Noctiphobia
From my experience, non fast-food restaurants generally accept credit cards.
As for musea, it depends, so bring always some cash with you, to make sure.
Some smaller shops
require[d]
some minimal sum (not sure if it's legal now), but it was so tiny (about 10 PLN) that you're unlikely to met that sum in a restaurant.
If the credit cards are accepted, those are practically always 2 main systems (Visa and Mastercard).
I've never seen any point that would accept one of them, but not the 2nd.
However, if you use more exotic system (like V-Pay), you should expect it won't be accepted in most places.
Please note that even if credit cards are accepted, it happens that the terminals are broken and in that case you won't be able to pay with your credit card).
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I think there's no 100% general answer, but card payments are so popular in Poland that I find it hard to imagine that any bigger museum in Warsaw wouldn't accept it.
The same goes for restaurants.
Regarding any minimum amount, it may happen, but it's more often the case in really small shops, and then this amount is really small (like 10-20 PLN) and you would rather always have bill to pay bigger than this minimum.
About the museum you mentioned, I checked their website and it doesn't state anything about credit cards, which I would interpret as "they don't say it's not accepted, so it is accepted", however they also recommend buying tickets online: http://www.polin.pl/en/planning-your-visit/admission
travel.stackexchange.com/users/7084/Kamil Mikolajczyk
I stayed in Poland (Warsaw and Poznan) for a month but never had an opportunity to use cash (I withdrew 600 PLN).
All the stores I went to accept my credit card, and it was very difficult to find any place that doesn't accept credit cards
(I only found one, which is a cheap bakery in Poznan and
it put a warning note on the entrance window saying "Cash Only").
As to the minimal sum, I didn't experience it, be it <3 PLN a bottled water in supermarkets, <4 bottled juice in 24H alcohol shop, or <9 a cup of coffee at cafes.
As to the brand, most stores only accept either Visa or MasterCard.
Amex is rarely accepted in cheap restaurants and cafes, except some chained stores (
Wedal, Starbucks, etc).
I also warn you that in Poland, it is more likely than not that you must choose whether you pay in PLN or your currency.
It is usually advisable to pay in PLN as the transaction cost is usually higher in your currency (but depending on your card).
Sometimes clerks choose your currency without asking you, but in that case you can ask them to switch back to PLN before entering your PIN.
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At this moment, being a resident of Poland
I find it impossible to find a cash only place at all.
Even the smallest vendors (food trucks etc.) accept cards.
I would recommend to keep 50 PLN for a traveller JUST in case.
But it looks like card payment penetration in Poland is well above 98%.
travel.stackexchange.com/users/86619/kotiformpl
Are credit cards accepted in the museums and restaurants in Warsaw?  When used in a restaurant, is there a minimum amount you have to spend in order to pay by credit card?  I'm particularly interested in whether the POLIN Museum of the History of Polish Jews accepts payment with a card.
Are credit cards widely accepted in Warsaw?
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<payment-cards><poland><warsaw>
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travel.stackexchange.com/users/60882/Nikhil
With the exception of some fast food restaurants, museums and almost every other establishment in Warsaw will accept credit cards beyond a tiny minimum amount (about 10 PLN). In Poland it seems to be very hard to find a place that accepts cash only. In most/all cases card is accepted and minimum spending amounts seem to be illegal so there should be no problem running into them in stores, restaurants and of course the Polin museum. 
Poland seems to be very good with card payments and as it seems to be a difficult place for your to find places that only accept cash. This includes the Polin museum.  Some shops may have a minimum purchase amount requirement that is present for card payment, however it seems to be illegal to do so now.