What the heck - it's Saturday, and my favorite show is on WUMB. Somebody more awake can check the arithmetic and assumptions.
Approximate latent heat of evaporation of water: 550 cal/g at 80C. (According to my ancient 10th ed. Lange's Handbook of Chemistry, it varies from 574@40 to 540@100.) So allowing one gram to evaporate from
100g will drop the temperature 550/100=5.5 degrees C. One gram is about the amount that will just wet the lid and pot. So pouring 80C water on a just-filled pot and letting it dry will (ignoring heat capacity of the pot; per below a reasonable approximation) drop the overall temperature by about five degrees.
Heat loss by radiation is nil (easily demonstrated by putting a hand next to and then above a full, hot pot, or solving the Stefan-Boltzmann radiation equation), and by passive convection only a little greater. So for you hot-pot fanatics, rinsing the outside of a previously dry pot may not be an optimal thermal-management strategy, though it's no doubt fun splashies.
Now to that other matter: step-drop from cold pot. A small sampling of at-hand Yixings in the 4-oz class gives a mean weight around 120g, and the specific heat of such artificial rocks is about 0.2. Assuming a starting water temperature of 80C and pot at 20C, ignoring leaf (as one does), I get the pot warming by about 48 degrees and the water cooling by 12, for an equilibrated temperature of 68.4 degrees.
Conclusion (for those who care about such matters): pre-warming the pot is GOOD (especially if you wipe it dry immediately); post-rinsing it is BAD (unless you keep it wet with hot water until steeping is done).
I hope that that excursion has helped to disobnubilate this perilous operation.
-DM