Not sure if it would generate any reasonable revenue, nor if it's actually viable in the presence of our current financial institutions. But I can't see any real downside.
But that's my point entirely: this requires an action, a decision on the part of the user, to press it. For the most part, I expect that people will nearly unconsciously, instinctively avoid pressing it. Part of this will be to avoid the "decision cost", part of it is the huge extra psychological weight imposed on actions that are (perceived as) free vs. non-free. (cf. app store free vs IAP mechanics, etc.) The very fact that the button requires an explicit user action and "costs something" dooms it.
This problem isn't a technological one. It's not "add button". It's "devise and bootstrap a distributed culture that agrees with human psychology and that aligns user and producer needs better than what we have today (i.e. ads, subscriptions, etc.)"
So do "Like" buttons, but people seem to have no issue clicking those.
I think that this might be helped if it acted like playlists, or bookmarks, where clicking the button would add the site to a list. Later you can opt to donate to items on the list (or provide a single donation split evenly among everything on the list).
Of course, I doubt many would actually use such a feature either.
It doesn't even have to be used by many people. Most people rarely click on ads, and yet a lot of websites are doing just fine with ad revenue. You just have to ensure that the small percentage of people who do click generate enough money to compensate for hundreds of others who don't.
However, I was replying to a comment about a "donate" button on the browser, and trying to keep within the problem that it is trying to solve, which is (at least as far as I can tell) paying for the vast number of far smaller sites that make up the internet.