Utter nonsense. I have always worn short-sleeved shirts, with, and without ties. Most offices in Britain have still to embrace air conditioning and I have always found them to be far too hot and stuffy, winter and summer. There no reason that a chap can't look smart with a short-sleeved shirt, other than old fashioned prejudice. Jeeves still lives it would...
why would you want to? A rolled up unrolled sleeve looks creased and tacky. Short sleeved shirts in the office, with or without a tie, are perfectly acceptable.
The point I am making is, with a long sleeved shirt, you have a choice, with a short sleeved one, you don't! I'm not advocating rolling up long sleeves!