Just dropping my $0.02 as a long time open source developer/maintainer, user of f-droid, and creator of an app that would be affected by this change (EteSync).
I’m not familiar with the original app that triggered this discussion, so I’ll just focus on the proposal regarding paid features in open source apps.
FLOSS is about user freedom, not costs as illustrated by the famous Stallman saying: “Think free as in free speech, not free beer”. Following that line of thought, any app on F-Droid that is not fully open source, that is, restricts your user freedoms should be marked as such, and guess what? They already are. It’s the class of “Non free X” anti-features.
A good example of this is Telegram: it’s non-free when in comes to libre (server is closed-source and network doesn’t support federation) but it is free when it comes to gratis. Is this any different to a competitor that would charge you $1? $0.01? Not in the context of F-Droid, because again, F-Droid is about FOSS, not about money. Both cases are equally bad.
Two good examples from the opposite point of view (and I’m sure there are many) are Conversations (as mentioned by @uniqx) and my very own EteSync. EteSync is completely FLOSS. Client and server. You can host your own and I actively help and encourage users to do so. Conversations is a free XMPP client that you can use with whatever server you want for free and no restrictions, but you can also use the XMPP server maintained by its creator.
In both cases the extra services provide additional value to users without restricting any of their freedoms. It’s a bit ridiculous to mark EteSync or Conversations with anti-features because they are trying to offer services that make it easier for users.
Also, in the case of EteSync, the “official” hosting is the only one available for public registration (that I’m aware of). How should we mark EteSync the app if it’s removed and another person, not associated with the project offers paid hosting, should the anti-feature be there?
On a personal note, I we should encourage the sustainability of open source software, not discourage it. The offering paid enterprise support (like in the case of Red Hat) or offering paid-hosting in addition to supporting self-hosting are great ways to do that. Let’s not ruin it with scaring users away or considering them as anti-features.