That's a good question. Unfortunately I don't have a solid answer for you.
Reading the tooltip, it sounds like they'd be on seperate cooldown, but I'm not certain.

If you'd like to test it, I'd recommend using a Burning Spell Weave set and a bow with lethal Arrow.
The former greatly increases your chance of applying the Burning status effect, and the latter always applies the Poisoned status effect.
The two together should get you a decent shot of getting both status effects to proc within a 5 second period so you can test.

However, I would caution you that if you plan of trying to make use of both for sustain, just remember that these status effects arent going to be easy to keep up under normal circumstances; the chance for a status effect to proc from a DoT(which is most of a DKs flame/poison damage effects) is quite low and really only procs periodically even when you're using multiple Dots. I.E. All flame damage morphs (Burning Embers, Engulfing Flames, Burning Talons, etc)
I fear that if you split the morphs, in addition to stat-splitting the scaling, that the chance for both procs would be reduced greatly.
At that point, it's quite unlikely in my opinion that both effects would proc anywhere near every five seconds anyway.
Just a word of caution, no offense intended. I have no idea how experienced you are with the game. Just going the extra mile in case you, or anyone who reads this, is a newbie

Good luck and have fun! And if you do end up testing it yourself, don't forget to post your results!