Can the produce flame cantrip be put out by weather?
I assumed that the answer was no, and that it's constantly being generated. But I'm unsure.
It cannot be put out by the weather or any environmental condition.
Jeremy Crawford's tweet about whether create bonfire works underwater:
Create bonfire lasts for its duration, regardless of the environment.
So, produce flame lasts for its duration (10 minutes), regardless of the environment.
It might be affected by weather, flickering, shrinking in size, but the flame stays and the spell is still active. The flame is maintained by magic.