Do Fire Elementals light the area around them?

[Swashbuckles]

Nothing in RAW, though the flavour text says it's like a mobile bonfire, so I'd normally rule it's at least as bright as a torch (maybe even more since it's a great deal more fire, even at small size).

Where I run into a grey area (pun perhaps intended) is what happens when that fire elemental enters the area of a Darkness spell. It took a 2nd level spell to summon (higher for bigger and arguably brighter elementals).

How should I treat this in my game?

PS: The elementals have darkvision, so even if they can't light way, they can see where to go, I guess!

[Azothath]

as you are asking in the Rules forum, we have to say technically No.

Sensibly, yes. Use Vision and Light as your guide. Water and Earth also can have glowy inclusions/fungi/algae. As long as it is descriptive you are good to go - just don't have it dazzle foes within 10ft, etc.

[Melkiador]

A lot of things about elementals don't make a lot of sense. For instance, you can grapple a fire elemental without getting burned.

They are described as resembling a bonfire, so I do think it'd be reasonable for them to provide that much light and warmth, but there's a hole in the rules there.