Version 4 of this license has pulled out the NonCommercial definition into a separate paragraph, which reads as follows:
NonCommercial means not primarily intended for or directed towards commercial advantage or monetary compensation. For purposes of this Public License, the exchange of the Licensed Material for other material subject to Copyright and Similar Rights by digital file-sharing or similar means is NonCommercial provided there is no payment of monetary compensation in connection with the exchange.
If your website will be "primarily intended for or directed towards [...] monetary compensation," then it violates this term. The next sentence clarifies that a barter exchange of one copyrighted work for another is not considered "commercial" in this context unless money changes hands. To my understanding, that sort of exchange is not unheard of in certain artistic communities, but it is not relevant to your question.
What this really comes down to, in your case, is how "primary" your donations would be. If you're running the site for its own sake, and donations are just meant to defray costs, then you might have a stronger legal leg to stand on (compared to e.g. running ads on the site and making a profit from it). If you incorporated a charitable organization or trust, that might also tend to help, but it must be emphasized that this provision contains no bright line carve-out for such legal entities. A trust or charity provides a legal framework for ensuring that donations are spent only on defraying costs and are not taken as profit, but the license doesn't automatically recognize that as "good enough." In fact, the license does not explicitly authorize defraying costs at all, but you could argue that the donations are just a means to the end of running the website (and not the "primary" purpose of the website). It is unclear to me whether that argument would work.
A judge would have to look at the totality of the circumstances to determine whether you are "primarily" running the website for monetary compensation, and it is difficult to predict how the judge would rule. Such a ruling might also depend on the jurisdiction in which the case arises. You should consult a lawyer as your question suggests.