I haven't had it "varnish" anything yet. This is mostly an Internet myth.
It may be mostly an internet myth but I have seen it in real life. When my father passed on, I had to manage his guns. He hadn't touched them in probably 10 years, or more. A couple of his guns were literally varnished shut. His Colt Govt model, stored in a dresser drawer, mag loaded, chamber empty. When I cleared the gun, the slide moves slooowly forward, stopping about half way shut.
Savage 99, varnished shut to the point were the safety lever could not be moved and the action could not be opened.
Liberal application of WD-40 dissolved the varnish (dried oil), allowing actions to be opened and the guns broken down for a proper cleaning. It takes some time, but with enough time, oil turns to sludge, and sludge turns to shellac.
Or something very closely resembling it.
As to oiling a gun until it is dripping? Some of the top pistol competitors from the 80s, & 90s have remarked on how they like to run their 1911s "wet", meaning not actually dripping oil, but almost. I've never run mine that way, but then I'm not a speedracer kind of shooter, either.