If I knew I'd had a tire fail during the takeoff, I wouldn't be attempting to retract the gear.
At that point, I don't know what's going on with the gear, what's hanging out, and what damage has been done. I don't know what damage may have occurred to the gear door, to tilt, to brakes, or to the surrounding structure.
Attempting to go cross country means raising the gear; I understand the sentiment that if the weight is going to be lost, it can either be dumped or burned, and why not burn it enroute, but it's not that simple, and it's an unnecessary risk to continue the flight. The other side of the coin is what may happen if one gets down the line, away from the possibility of an immediate landing, and something cascades, or the gear gets stuck in the well because something is hanging out, or there's damage such as a fuel leak from an exploded tire (concorde, anyone?)...imagine getting there and wishing one had simply returned to land.
I'm of the thought, not knowing full circumstances and only faced with a theoretical per the thread question, of leaving the gear in place and returning to land at Newark.