What is the range of the EOD bot?

I've recently unlocked the EOD bot, and while playing around with it (and being hopelessly ineffectual with it) I've noticed that after I have driven a certain distance away I will return back to first-person view. Running towards the EOD bot will allow me to take control of it again.

How far can I drive an EOD bot away from me before I lose control of it?

I started on an empty server playing Operation Firestorm (conquest large) with a 1 player start, and drove the EOD bot all the way from the US deployment to the RU deployment, and beyond -- to the very edge of the map on the opposite side.

Based on this, I am confident that the range of the EOD bot is effectively infinite, or at least as large as the very largest Battlefield 3 map so far.

So, I'm not sure why you're losing control of the bot, but range is not the reason.