It's a fair assumption, but...
The short answer is no, rain does not harm them.
No guesses, but we can examine the evidence if there is any ;)
I haven't seen it rain in the Ashlands.
The Surtling presence in the Swamps, where it can and does rain might be explained by the protection that the Fire Geysers provide.
Have you ever seen the Surtlings travel outside of the Ashlands other than a raid event?
When a "The smell is fading" event occurs, has it ever rained?
If and when that raid event occurs and it rains, then you'll have your proof one way or another...
Alternatively, you may accept the information the Wiki provides ;)

Originally posted by Wiki:

They are especially dangerous in groups, but are weak to Water and receive damage from it, although rain doesn't seem to bother them.

https://valheim.fandom.com/wiki/Surtling