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DPS heirlooms don't work very well when tanking, so if you choose the "level as a tank solely through dungeons" route, you'll want to pick up applicable gear as you go. You'll lose the leveling bonus, but you won't risk getting kicked every time you queue because you're wearing the wrong items and therefore possibly having trouble as a tank.
If you want the full benefit of your heirlooms, I'd say go Arms & quest while in the queue.
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