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All the different DK fire DoTs stack. Its how DKs end up with such huge sustainable DPS numbers.
EDIT: This should just say all the different fire DoTs stack, not just DK abilities.
The fire enchant does not stack with the burning effect from inferno staves, therefore the enchant on the staves should be poison to ensure an extra DoT.
OP, you should be applying Engulfing Flames and Burning Embers (if you're magicka) consistently to keep a DoT rotation up, they do, indeed, stack, as others have said previously in this post. (:
Unfortunate magicka warden main. PC/NA Server Fairweather Friends Retired to baby bgs forever. Leave me alone.