The real answer is, it depends.
The biggest factor is the design of the all wheel drive or four wheel drive system and what type of transfer case it uses.
Some all wheel drive systems use viscous couplings and other limited-slip designs with torque splits, that are sensitive to different rotating speeds front to rear. The problem is, if the difference is too high, these heat up and wear prematurely, causing them to fail. These are mostly found on cars, though, not trucks, and there will always be at least some acceptable level of difference, as the car naturally experiences differences in speed when turning or when the weight load changes, compressing the sidewalls of the tires.
Pickup trucks and full size SUVs tend to use traditional four wheel drive systems with a locked ratio transfer case with a direct chain drive or similar, and locking wheel hubs. These four wheel drive systems will generally be fine with significant differences in front to rear wheel speed. The problem you'll run into running different size tires with them is excessive tire wear, since the tire is the weak point in the system instead of the transfer case. You may also need to change the gear oil sooner than normal, but it shouldn't cause total failure.
Find out the manufacturer guidelines for your specific all wheel drive system, and follow them.