Beergun bottling question

Is it safe or ok to store beer that has been bottled with a beergun at room temp or does it need to be kept cold for long storage?

I don’t use a Beergun, rather I use a modified cobra tap. But the result is the same, so having said that, I can say from experience that if you’ve been careful about sanitation, oxidation, and used PureSeal oxygen scavenging caps, you can indeed store at room or cellar temperature. I have even opened bottles after more than 5 years and tasted no ‘off’ flavors whatsoever.
Of course, how long a beer will last at room/cellar temps will depend on the beer. A Barleywine/Burton ale will keep quite well (and the last bottle you consume will usually be the best one); on the other hand, a cream ale, ordinary bitter or a mild is not going to do quite as well and are best consumed soner than later.
And whatever you’re bottling with the gun especially stronger beers or something you think might be sitting at room temps for more than a few months…you’ll also want to make sure that all residual fermentation is truly finished before you condition and bottle the beer.