If your shield has an 'Inflammable' prefix, you won't catch fire and take damage from burning, even from the Hellborn skill tree.
It seems to be corroborated by these various posts from the Gearbox Software Borderlands 2 forums:
Don't use anything labeled 'Inflammable' you will not catch fire.
Source: Chance to burn vs. resists?
Inflammable cancels it all out, but a shield that reduces fire damage should still work.
Source: Hellborn and Shields
I did have the 'Inflammable' but that renders most of that tree useless if I can't set myself on fire.
Source: I Hate Burning
Q: Can you still light yourself on fire with an inflammable shield? A: Tested and no. Wouldn't be enough of a "risk/reward" system if you could.
Source: Can you still light yourself on fire with an inflammable shield?
...from what limited testing I've been able to do so far, inflammable shields prevent you from getting the bonuses associated with being on fire.
Source: Inflammable Shields and Hellborn
Inflammable does not work, since you can't get lit on fire.
No, it won't work, inflammable mod nullifies the self ignite thus making the hellborn skills bonuses not work.
Source: Does the 'Inflammable' prefix work with Hellborn?
From my playing with an inflammable shield I'd have to say it prevents you from ever self igniting.
Source: Inflammable shields hindering Hellborn?