Yes, the bushing will reduce in ID and create resistance on the shaft. The clearance must be large enough to accommodate this. The clearance also must be large enough to accommodate swelling due to water or lubricant absorption.

Use the coefficient of thermal expansion to calculate the change in volume of the bushing. Assume no change in OD or length. All of the increased volume will result in ID reduction.

If the large, rigid body has constant temperature then the bushing o.d. will have that temperature also - probably not going to be the case, though. Is the bushing a fabricated piece, or is it purchased? Bushing manufacturers publish guidelines on installation bore and shaft diameters.

A metallic bushing with high thermal conductivity will operate at a very similar temperature to the housing. A plastic bushing with low thermal conductivity is going to have a much higher temperature gradient across the bushing. Assuming it's all hot is a safe design consideration. If a smaller clearance is required then a thinner walled bushing should be considered.