Do hot drinks cool you down?
A Study by Lee &amp; Shirreffs addresses this question.  Unfortunately I only have the abstract, but it's enough to summarise from:

Method


Three groups exercised by  cycling before being given water at either 10°C, 37°C or 50°C.
They then continued to exercise until exhausted. 
Mean skin temperature, core body temperature and heart rate were measured throughout.


Results


Mean skin temperature was highest in the hot drink category - half a degree warmer than in the cold group.  
There was a heat difference between the two outlying groups of 33kJ.  


Irritatingly the pre-drink temperatures are not included in the abstract so it's hard to compare the effect that the drinks have had.  Certainly the hot drink has warmed skin temperature above the thermoneutral group but I suppose that could be due to increased vasodilation leading to better core heat dissipation (as you suggest).  

In absolute terms, you are warmer than you would have been after having a neutral or cold drink, however it could be that you feel cooler as your hotter skin loses heat energy at an increased rate due to the greater gradient.  



References


 Lee JKW, Shirreffs SM. 2007. The influence of drink temperature on thermoregulatory responses during prolonged exercise in a moderate environment. Journal of Sports Sciences 25: 975–985.