Will it damage my 4x4 if I install new tires on the rear only? All tires are factory oem size.

Only if you drive it in 4wd on a hard surface, the driveline will tolerate a little difference in tire diameter like new tread versus worn tread on the same size tire but that would be about it. I would not drive at higher speeds in 4wd with odd tire size either.

Even with same size, identical tires, I can't drive on hard surfaces & turn. I once was on a soft lawn, had 4WD engaged, so as to not tear up the Town Green. Pulled onto paved lot & started to turn... couldn't get it out of 4HI. The truck hopped like a bunny trying to releave the pressure. I had a battle before I got it disengaged.

I would put the new tires on the front end. That way you won't have as much wear on them when you replace the rears. If I buy 2 tires, I always put the new ones on the front and the older, better ones on the back. This way there is less chances of a blow out and then causing a wreck.

I know this isn't exactly a right on the topic but... With all the steep grades and washboarded roads out here I learned a few thing about 4WD Dodge Cummins trucks.1. You can use 4WD going up a washboarded road just be aware to swing wide to keep the front axle from binding up.2. Do not use 4WD going downhill with a exhaust brake. What happens is all the engine weight over the front axle, bouncing up and down from the washboard and then the braking force of the exhaust brake will break u-joints even solid u-joints.So as for the mismatch tires I would be careful on any dry solid road even dirt. Now in mud and snow you'll be fine because the tires can slip a bit.

Thank's for the advice, what I want to know is if I put new tires on the Front or Rear will I damage my truck if I drive without using the 4 wheel drive. Drive in 2 wheel drive only?

Just 2WD there will be zero effect. Just if you use 4WD just be aware of the difference and keep a feel for the binding. You'll know...