I will preface this question by saying that the story in question is anecdotal, told by an instructor at an airfield I was visiting. I will obfuscate the callsigns and airfield in question.
Apparently, a pilot got onto the frequency and said: "[Airfield name] Traffic, [callsign]"
[Airfield name] ATC responded with: "I've been to University for 8 years and have several million pounds of equipment here, I'm not just some guy with a walkie talkie, please turn around"
He proceeded to leave the airspace and diverted to a different airfield, however, if he didn't, what would be the potential repercussions? I'll add this was a VFR flight with no flight plan.