At my university we are all required to wear long sleeve coats in chemistry labs. It simply protects more of your body compared to a short sleeve coat.
As an example, in one of my labs a guy had his sleeves rolled up and then he decided to rest his forearm on the fumehood while he was working with sulfuric acid. It turned out he spilled some acid earlier so he ended up with a burn all along his arm. This could also happen if pouring chemicals - if you spill some, your arm won't be covered.
Long sleeves are also important when handling liquid nitrogen with cryogenic gloves. You need to have your sleeves going over the ends of the gloves, to prevent any liquid nitrogen falling into the gloves where it can't evaporate easily off the skin and could cause burns.
I suppose one hazard with long sleeves is that spilling a chemical on your hands could cause it to roll down your sleeves and increase the contact time of the chemical with your skin as you then need to take the time to remove your lab coat before you wash your arm, but ideally your coat will be tight at the cuffs to minimise this risk.