They draw the blood under very clean conditions (at least, they should...), so the risk of infection is very low. Once your blood clots off and the wound seals itself, the chances of it getting infected from an outside source are very slim. I would say that that takes no more than an hour or so. So basically, they put the bandaid on your arm and you leave it there for a while and by the time you take it off you probably aren't going to get it infected unless you scratch at it a lot and rub dirt in it or something.
Most likely, you are just having some tenderness because somebody stuck a sharp metal object through your skin. Go ahead and ignore it unless it gets red, hot, you see red streaks going up your arm, or you get a fever.