I think this is a deeply, deeply personal question. I think, for me, stress comes with responsibility. So in that case, my knee-jerk reaction would be to say tanking is on a par with healing for responsibility. If either the tank or the healer isn't doing their job properly, the party will likely wipe. But a bad tank can be mitigated by a great healer, while a bad healer can only be mitigated to some extent by a great tank, in my experience at least.
But let's not forget the old adage that if you're struggling to down a raid boss, you need to drop either a healer or a tank in favor of more DPSers. DPS kills bosses; at the end of the day, you can't heal or tank a boss to death. And DPS performance is by far the most measurable and visible. Sure, there are healing meters, but any raid leader worth their salt knows that you shouldn't base your opinions of a healer solely on their raw throughput. There are too many other factors involved. And tanking isn't really measurable on meters -- that I'm aware of, anyhow!
DPS, on the other hand, is there for all to see. DPSers are putting their skill up for all to see every time a boss is engaged, and even if a fight goes well, lower DPS could still be criticized. People rarely criticize tanks or healers unless there's a wipe. Then it's all their fault.
And looking at PvP, I think all the roles are stressful, but maybe healing is the worst. You're the focus of attacks far more regularly than other roles. Not only that, but you have no control over your fate. You have to rely on your teammates to kill people, although of course you can assist in many more ways that simply emptying your blue bar into their green bar.
Personally, I do find both tanking and healing stressful, but for me that's good. I like the game to be engaging, fraught in places, pushing me. I have to admit, I find PvE DPS a bit dull. I'm not in control of my survivability very much and largely have to hand that over to the healers and tanks. All I do is execute my rotation and try to smack the boss harder than the guy next to me. PvP DPS is a whole different story!
But that's just me. Tell me about your thoughts. Which role is the most stressful? Are there roles you wouldn't do because they're too stressful? Which role is like a walk in the park?