I had a e8200@e8400 speeds and well singleplayer was playable but not to enjoyable and well multiplayer just sucked.
I wouldn't even bother playing it online without at least a core i3 2100 at bare minimum,i upgraded to the i3 2100 at one point from that e8200,it made a tremendous difference and its my bare recommendation on a budget if you want a good BF3 experience.
The newer i3 series would be better sought after but used and on a budget,a h61+i3 would be worlds better over that core 2 duo,i know from experience and match it with a 6870/7700/gtx460/gtx560 and its a well balanced system.