The reason shaman's can't use Ankh's or their Elemental pets in the arena is due to the fact that they both have a 30 minute or more cooldown. Heart of the Phoenix is on a 10 minute cooldown, I see no reason why this wouldn't work in an arena setting. Although speaking from Beta experience, your pet does come back with 10% life, and fairly quickly I might add.
In fact, when you FD, your pet just sits there and dies. It takes all of 1 second to resurrect, and get killed again by the enemy. There's no wait time between the time that the pet is killed and the mob de-aggro's, so it gets killed twice. To be honest it's rather annoying at the moment.
So taking this into account on the arena side. Your pet, who has MAYBE 10K health with new arena gear, as well as talents (big MAYBE), will receive 10% of their life back when they resurrect. So that's at most 1K health, which anyone knows is just waiting to be killed in an arena setting. Currently this talent only works for Ferocity talent tree's. You may want to bring a bird, raptor, or even a crab with you for pvp settings (look up their new abilities), but only time can tell that for now.
We've already given out the suggestion of reworking the talent to possibly be delayed, or a talent you could personally activate instead of it being instantaneous. We'll see what the developers come up with.