Enduring the Heat

Best route in my opinion: (use runes as you come across them, of course)
From the entrance, go straight forward until the cavern wall, follow it to the thermal vent, jump across, to the next thermal vent, keep left and follow the wall, then turn right, stamp the rune in the middle and turn left, to the wall again and follow it left to the last rune.
You should be able to just ignore the spawning elementals, they are destroyed when they are close when you douse a rune, and all remaining ones as well when you complete the last rune.
With this route, I rarely stay in combat after leaving the cave again, probably because you can get out so quickly from that point.

If you have trouble surviving (which is likely due to other players doing the quest at the same time, spawning lots of elementals) simply stand in a rune, this will grant the buff [spell=97764]*, reducing fire damage taken by 90%. As the elementals only use [spell=97884], which is fire damage, you only get around 50 damage per melee hit. Wait for a bunch of elementals to gather, then destroy the rune and continue your course.

* Yes, the quest text indeed mentions this:

You can use these runes to protect you from the fiery creatures [...]

For those like me who were wondering how to "stamp" a rune, you do not hop on it. Mouse over it and right-click. *Edit: Icy Veins made a nice picture of the route to take here. They also has a guide for all the dailies here, as a bonus.

Allow me to state the obvious here, because somebody has to:

DON'T STOP. KEEP MOVING.

Unstable Flameragers will keep spawning as long as you're in the cave. Unless you overgear this quest, you will not be able to kill them fast enough by yourself. Keep moving from Rune to Rune -- detonating a rune will kill the Unstable Flameragers that are within range of you.