In my experience with Rams, you can usually go up approximately 1" in tire size from stock/OEM without rubbing. Any more than that, and you tend to get some rubbing in turns.
275/65R20s are basically a 34" tire. What options do you have besides it being a Longhorn? If you add that, or add a signature with that info, it will help us.
For the 5th gens, they should fit fine, according to what I read.
If you have the off road group (ORG), you have roughly an inch more height than a "regular" Ram. 34" tires should be fine there, I believe.
If you do not have the ORG, and you will probably experience some rubbing on some of the plastic trim when doing turns, unless you're in Off Road 1 mode (if you have the air suspension).