The DM should have the ultimate say as to what races and classes are available in the game.
Now, Mystara is an interesting case as it was the setting for classic D&D, and those rules had fewer options in core than AD&D, plus races were their own class. Core D&D didn't even have gnome PCs, though there were rules for typical gnomes in Top Ballista. But Mystara was also a kitchen sink, so like Greyhawk and the Realms, it's pretty flexible too. Half-elves and half-orcs aren't that much of a stretch. Drow weren't in the setting, because they were an AD&D thing, Mystara had the shadow elves instead.
Tieflings are a possibility, but they should be the traditional ones, not the homogenized 4e ones. Devils weren't traditionally used in Mystara, so the 4e tiefling backstory is at best an awkward fit. Mystara does however have the 7 original demon types in the Immortal rules, as well as Demogorgon and Orcus, so the old school Planescape tieflings wouldn't be a problem, but they should be rare. They'd probably face a lot of prejudice in a place like back country Karameikos, but on the Savage Coast, they could probably be assumed to be under the Red Curse. But that would lead to its own flavor of persecution, and if they hang around long enough, they'd probably fall under the curse as well....
Dragonborn don't really fit in the Known World or other explored parts of the world, but there's still lots of unexplored territory in Brun and Davania, and Skothar has never been officially described. So I'd limit any dragonborn to those areas. Any PCs would probably be additions to a party exploring the unknown reaches of the world like the adventures on the Princess Ark, and probably shouldn't be at the start of the campaign. Or maybe dragonborn are extinct on the surface and can only be found in the Hollow World.
That's just my take on it though. I do fully agree that a DM is well within his rights to disallow nonstandard races, and in 5e, that's anything not human, elf, dwarf, or halfling. Everything else in the PHB is there for backwards compatibility.