I'm assuming by 'open track racing' you mean HPDEs. And no, most organizations require long sleeves, however at my last SCMC/TrackClub USA event, they allowed short sleeves.
^ +1, most HPDE's require long sleeve cotton shirts and pants. I DO recommend a fire jacket though, nice to have in case something goes REALLY wrong, but hopefully it will never be put to use. I have a simpson jacket, it's comfortable and breathes well.
Might as well just buy the full hero suit aka driving suit, the jacket wont cut it when and if you make the move to wheel to wheel racing. Running just a coat is more for peace of mind but it leaves your legs S.O.L when and if you need fire protection.
For HPDEs and most Time Trial racing, long sleeves and long pants. Generally, the decision to allow short sleeves is up to the track personnel depending upon weather.
While running hillclimbs I use Jacket/Pants instead of a suit. Being on the top end of the large scale, I find it easier to move around and more comfy in those, rather then a suit.
However when you get serious it is a good idea to run a full Nomex suit, gloves and shoes.
You guys remember this shot of Joe Johnson bailing out of his R when his vapor valve failed and his dumps heated up and pressurized his cell?
I wear Nomex coveralls, socks, gloves and balaclava when I opentrack. This allows me to wear shorts and a t-shirt when I'm sitting around waiting for the next run.