Most players will use that sleeve to keep their arm warm or to hide an injury. Sometimes a sleeve might me really torn and is better off removed. And some players just do it as a fashion statement.
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If football offensive linemen, wore long sleeves it would give their opponents something extra to grab a hold of to throw them out of the way. Additionally, offensive linemen have their jersey sleeves tied up instead of hanging loose like their other teammates.
the quarterback wears on sleeve so that while they are on the sidelines their throwing arm and shoulder stays warm to prevent injury
1. To help against turf burns and 2. To help against the elements.
Sometimes even to help with ball control.
they dont wear long sleeves because they are running around and not getting cold