Does Grinning Axe need the OP ring?

[Carlos]

It already has both Burn and Ignite, would the Ring of Infernal Devotion help? (the one that causes Burn weapons simultaneously cause Ignite). Like would it enhance the effect somehow, inflict the status faster?

[Fried Noodles]

It's not strictly needed, but it does help.
You want the ignite to be triggering as often as possible.

It's easy enough to test too. Find yourself a convenient slightly-tanky enemy and give 'em a few whacks without the ring, and a few whacks with the ring.

If you can save a whack or two, then it might be worth a ring slot.