It could influence new viewers initial opinions. The site having 5K subscribers instead of a handful...

Actually, according to Aweber, it does (at least for email subscriptions, displaying a chiclet with subscriber numbers increases opt-in).

It's a form of social proof.

Is there some sort of evidence of that? I guessed no because I assumed most people don't even see it, but maybe people on the fence about subscribing do.

There's plenty of evidence that social proof works, it's just a questions of whether there's something special about FeedBurner that would invalidate social proof mechanics.