Continual flame does not produce heat nor require oxygen and lasts until dispelled:
A flame, equivalent in brightness to a torch, springs forth from an object that you touch. The effect looks like a regular flame, but it creates no heat and doesn't use oxygen. A continual flame can be covered or hidden but not smothered or quenched.
If it doesn't produce heat, it presumably does not burn whatever it is touching.
Now suppose you have an owl familiar. If you cast Continual Flame on the owl, will the owl become a flying torch? And if you dismiss the owl and have it reappear, does the continual flame, having never been dispelled, reappear as well?