Does the heat increase the amount of calories burned in a session?

Person B arrives at the station in 4 hours?

I wish cause the heat does suck and would make it slightly better

They will sweat more and lose more water weight but calories burned should be equal.

I would guess person A would be less sweaty.

No, you will not burn more calories in the heat. You will sweat more and lose more weight (water weight), but when you rehydrate, you will gain that water weight back.

Actually, you would burn more calories in the cold because your body will try harder to keep your body warm.

I've always heard that the body burns more calories in cooler temperature, because it is trying to warm your muscles up.

Is the wind blowing? Is it raining?

No

There is zero benefit calorie wise to running in the heat.

I've always heard that the person that runs in frigid weather burns the most calories bc his body has to consume more to warm to acceptable core temp.

My understanding is that energy used (calories burned) is measured in terms of heat equivalences needed to raise the temperatue of a standard amount of water. Of course in the human body that is not what the calories are being used for, unless the ambient temp is very low, in which case you would burn calories to heat the body. The process of cooling the body by perspiration flow and evaporation would not seem to make high energy demands on blood sugars or stored fats. Further, in the presence of body heating through exposure to high external temps, the blood vessels would expand and generate lowered pressure within the vascular system and therefore less work for the heart to supply blood to the muscles. Just intuitively, less demands on the muscular and vascular systems, less calories burned.
Worse yet, after a high temperature workout, expect a greater propensity to self-medicate with icy high calorie and carb beers.