If you have the roster spots and can carry both of them without using them in your active lineup, I'd hold onto them. I personally hung onto Konerko until a few days ago, but dropped him because I needed use of that roster spot with a bunch of injuries to cover. Since his slump has made him so low on the rankings screens, noone is likely to notice him, but I will still monitor his performance and when he shows signs of coming to life get him back.
Konerko has had three consistent seasons of .280-.300 averages, 30 homers and 80-100 RBIs. At 27, I just don't buy that he has lost it completely any more than Giambi or Tejada have lost it and he should rebound, probably as soon as Jerry Manuel is put out to pasture. At his age and with him apparently healthy, there is just no reasonable explanation (like new team, new ballpark, new league, etc) to explain just suddenly forgetting how to hit. If you need use of that space, I would dump Konerko but carefully see if there are any signs of him coming to life. He has always been an extremely streaky hitter; last year he hit 16 of his 30 homers in a one month span. Just get him back when he starts to turn it on.
Burrell might be a little more worrisome, because unlike Konerko, he doesn't have a track record of consistent performance. He had a major spike in power and RBI production as well as average last year, and had generally been improving every year, but only really had his breakout in 2002. Perhaps he can be compared to Aramis Ramirez, who went .300-35-110 in 2001 virtually out of nowhere and then hit only .233 in 2002 with greatly reduced power. With Burrell, there isn't really a consistent track record to show that he should be at least at a certain performance level, so last year could have been a career year.