My understanding is that 冷える describes the state of going from room temperature to a lower one, e.g. to chill beer, to get cold (weather, relationship), etc. On the other hand, 冷める is about going from a high temperature to room temperature.
If the above holds true, how come this sentence found under a dictionary entry defies such definition:
(Entry "温める," taken from ルミナス、研究社)
I do not think they are referring to gazpacho (served cold) and the like, because it does not make much sense to heat up something you intentionally let chill, that which is meant to be cold in the first place. Neither do I think it is a mere typo, since I believe a mistake as simple as such would have been edited out in a published dictionary.
*Much to my dismay, found under their 冷める entry: