So "Insists it be renamed" vs "insists it is renamed".

IMHO the first option sounds formal and archaic. The second sounds more in tune with contemporary English.

That said "insists they rename it" would get rid of the passive and avoid the problem entirely.

Yeah that was the first thing I noticed as well. I'm somewhat surprised (a little sad, even) that the BBC chose not to write it in the subjunctive mood, which is precisely what is required here.

I thought when I read the headline that Instagram was insisting (that is, reiterating) that Littergram was already renamed.

The subjunctive is ingrained in my internal English grammar model, as is the who-whom distinction. I've been thinking about dropping them, though. They have started to sound a bit elitist.

Well, you can always adapt your speech to your audience. But I don't see why it should be seen as elitist.

> in tune with contemporary English

Contemporary American English, sure. The former form is still quite popular in British English

I'm British. So is the BBC.

Bloody limeys, think they have English ...

Correction: "thinks they talk the English"