Taking the opposite view first,
Anything you take into your body that is colder than iteslf, has to be warmed to the same temperature, using calories. Also assuming that you are purifying your water by chemical means, you'll be consuming a finite resource and would eventually run out of chemical.
Warm drinks would not need to be heated by the body, but you'll have to invest the calories in harvesting fuel, though this would achieve the purification stage as well as heating the drink. Your fuel, harvested sensibly will be a renewable resource.
One thing that often goes unmentioned is that folk like to warm their hands on the sides of a hot cup - this is not always the best thing to do as your body will then be pumping blood to your cold hands and hence risk cooling the core even further - though this often goes without consequence in our centrally heated world.
Thankfully, I've never been in a "Survival" situation, but if I ever do, I'll do my utmost to create fire and I'll use it in as many ways as I can to keep me alive - one of which is likely to be hot drinks.
It might be that warm drinks don't actually warm you up, but cold drinks will certainly cool you down.
ATB
Ogri the trog