Two questions really: 1. Why does putting a spoon in boiling tea bring the froth down? 2. Also, the tip of spoon (outside tea) never heats up. I cannot imagine that normal air can have so much of a cooling effect. The heat conduction from spoon inside tea should be far more than cooling effect of air. Isnt it?
Taking the more Physics related question: if your teaspoon is made from stainless steel it has a considerably lower thermal conductivity than mild steel or (if you're at a swish tea party) silver. Depending on how much of the teaspoon is out of the tea it's entirely reasonable for the tip of the spoon to stay reasonably cool.
Your first question is more Physical Chemistry than Physics. I don't know the answer, but if inserting a spoon breaks the foam it's either because the metal surface acts as a nucleation point for the bubble walls to collapse, or possibly whatever surface active species is stabilising the foam adsorbs onto the metal rather than at the air water interface and this collapses the bubble walls. I don't think there are any strongly surface active species in tea, so the foam is probably easy to collapse.