Every instance of the Doctor meeting his past/future selves in the original series - the Three Doctors, Five Doctors and Two Doctors incidents - were deeply meshed in Time Lord technology. In the first instance, they nearly drained their own power supply keeping the three Doctors stable. Even the fourth incident - Time Crash - was the result of the TARDIS' shields being down. There are a few later examples, which I think are all Eleven running around in time loops and bumping into himself (off the top of my head - The Time of Angels, Time/Space, and Night and the Doctor all did this one way or another).

More generally, the Blinovitch Limitation Effect comes into play. Specifically, this version:

...it is a physical effect that occurs when two versions of the same person from different time periods make physical contact. This results in an energy discharge, shorting out the "time differential" between them. The Mawdryn Undead storyline establishes that the younger version of the character involved in the discharge, Brigadier Lethbridge-Stewart, is traumatized by the event and, for the next several years, loses his memory of the Doctor.

This has been inconsistently applied over the run of the series: in the new version, Rose held herself as a baby and summoned the Reapers, but Amy Pond has come into contact with her younger self numerous times with no ill effects. It's all a bit wibbly-wobbly-timey-wimey.

That said, OFUs are (and always have been) free to play fast and loose with the canon in order to get the staff together - OFUM has Morgoth and Sauron being painted urple by Merry and Pippin, which is just a bit unlikely, don't you think? They're humour stories, so like the PPC, it's all in the name of Funny.

hS

I actually had thought about the various counterparts or different characters. Davros verses Davros would be hilarious and Ten and his human self would probably argue over Rose. And there's Eleven and his ganger. Oh dear, One is going to wonder how all this happened and how his incarnations have an identity crisis.

It's not as though sudden portals that may or may not shoot pointy crystal are any more dangerous than keeping several known omnicidal maniacs on staff. Plus, the Cybermen would probably ambush students in their sleep to put their brains in robot bodies. OFU life is inherently a risk to health and safety. It's what learning through pain is all about!