Do airports charge airlines for extra time spent on ground due to weather conditions?
The correct answer may differ for each pair (airline, airport) and can be found in the contract that they mutually agreed upon. Such contracts are mostly seen as a business secret, and most of their contents are not known to the general public.

As far as I know, both parties agree to do their best to keep everything working during every weather. Which means that there has to be enough equipment to clear a runway really quick, such that these "penalties" are "background noise" in the overall bill, even compared to the penalties caused by other reasons. The airport may decide not to bill the airline most of these penalties, as long as the airline won't try to bill every loss caused by the airport's fault. As a rule of thumb, you don't upset your business partners like that over peanuts, more so if claims go both ways. They may even have agreed on how many minutes delay won't be billed.

Yet, this partnership may be put to the test when one of the partners is seen as cutting cost at the expense of the other, or when one really unexpected condition occurs (like that volcano on Iceland, which disrupted air traffic in central Europe for two weeks or more).