It shouldn't be any catastrophic stress from running in front wheel drive for a while. Now, I am assuming that you have the solid front axle - if it is IFS, you may have a lot of wear on the CV joints from the over use - but I would think that an F250 would have IFS.
It will handle strangely, but you can drive it around in front wheel drive as long as you have removed the rear drive line and it doesn't leak gear oil out of the rear slip yoke. I don't know which T case you have, if it is a bolted in yoke or if the slip yoke is in the case. If the yoke is fixed, no problem. If not, then you have a very good chance of loosing all your lube, which will trash your transfer case in quick order.
I've driven my Jeep around in front wheel drive for many a mile when the rear driveline shattered. It took several days for the repair, and I had no other car. But, big warning here - don't go too fast, because the vehicle will not drive/handle well.