What's the relationship between 暖, 爱 and 愛?
The right part of 暖(nuǎn) is 爰(yuán) and it's not 爱 or the traditional form 愛.

In ancient Chinese, 爰 is a kind of pronoun, like "where/which". And it also has other meanings, one is "to change". But if you saw "爰爰" in ancient articles it meant "leisurely".

Technically, there is few relationship between 暖（or 爰）/爱(愛). Or one can say by force, they have the same Chinese character component, the first four strokes on top of 爰/爱. It means 爪(tail), and in ancient Chinese it also means "hand". See? "to change" one should use his hand, and to love one might cook/make a gift by his hand.

You must distinguish 暖 and 暧(曖, ài). For 爰 there is no traditional form. 暖 means "warm/to warm", and 暧 means dim.

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㬊(huǎn) is not 暖(nuǎn) even one puts 日 over 爰, it is the opposite of 暧(dim), "bright". 
暖(nuǎn) has another pronunciation, xuān. "暖姝" （xuān shū）means "have a high opinion of oneself". BUT "暧昧" (ài mèi) means "ambiguous/dubious/shady".
暧 also means "warmly", so it is 暖... (Of cause this is contentious!)