「未読が1件、新しいトピックが4件残っています。または、他のトピックを...で閲覧する」

I have a feeling that the first verb here should be “are” rather than “is”. Neither look perfect in the current sentence (which is a bit of a muddle).

I think you’re right in this case. I’d use “are.” But I give a pass to things like this in interfaces – I imagine there are always more urgent things to code than perfect grammatic logic for the various possible combinations.