The job was probably not cursed.
There is no true evidence that Voldemort ever performed such a curse. No one saw him do it. No one heard him do it. No one heard him talk about it. The only piece of evidence ever proffered was the fact that no one held the job for more than a year after Voldemort was denied it. This is mentioned by Harry in Chapter Eight of Half-Blood Prince:
"That job's jinxed. No ones lasted more than a year... Quirrell actually died doing it... Personally, I'm going to keep my fingers crossed for another death..."
And by Dumbledore in Chapter Twenty of Half-Blood Prince:
"Oh, he definitely wanted the Defense Against the Dark Arts job," said Dumbledore. "The aftermath of our little meeting proved that. You see, we have never been able to keep a Defense Against the Dark Arts teacher for longer than a year since I refused the post to Lord Voldemort."
Unfortunately, though, neither Dumbledore nor Harry were statisticians. They apparently forgot the fundamental rule of Post hoc ergo propter hoc — the fact that following Voldemort's failure to get the job no one ever lasted a year does not demonstrate that because of Voldemort's failure no one ever lasted a year. It simply shows a correlation, and a correlation could be causative but it could also not be causative. Of course, with every additional year that goes by and another professor leaves the correlation is strengthened; however, even then it wouldn't demonstrate that the cause was Voldemort not getting the job. There might have been another event at around the same time that was the cause.
Additionally, the correlation would only be significant if it couldn't be explained perfectly normally without the existence of the jinx. Now we don't know most of the teachers during the relevant time period, but at least for the ones we see it is not surprising that they didn't last. Working backwards, Carrow was illegitimately appointed and ended up on the losing side of a war, things which would tend to make your term short. Snape was also only expected to teach for one year; at that point he would either be a criminal on the run or Voldemort's right hand man (depending on who would be in power at the time). Umbridge was only appointed in the first place because no other candidates could be found. She had no teaching qualifications and it was obvious from the beginning that as soon as Dumbledore would have the ability to get rid of her he would. Moody/Crouch was only hired for one year to begin with. Lupin was a werewolf. It was kind of obvious that once his secret would get out he would likely be forced to resign. Lockhart was an incompetent teacher. If you believe JK Rowling, Dumbledore hired him specifically to expose him. He clearly was planning on only having Lockhart teach for a limited period of time. If you don't believe Rowling, it is still eminently reasonable that an incompetent teacher wouldn't last more than a year.
Then we get to Quirrell. If your goal for the year is to steal the most heavily guarded object from right under the headmaster's nose, plus let a troll loose in the school and try to kill one of the students, it is quite likely you won't last that long. If anything the question should be how he did last the entire year. (Answer: [Dumbledore]'s a genius! Best wizard in the world! But he is a bit mad, yes.) More importantly, though, Quirrell is actually evidence that the job was not cursed.
When we are first introduced to Quirrell in Philosopher's Stone it seems pretty clear that he has already been teaching Defense Against the Dark Arts prior to Harry's first year:
"Professor Quirrell!" said Hagrid. "Harry, Professor Quirrell will be one of your teachers at Hogwarts."
"P-P-Potter," stammered Professor Quirrell, grasping Harry's hand, "c-can't t-tell you how p-pleased I am to meet you."
"What sort of magic do you teach, Professor Quirrell?"
"D-Defense Against the D-D-Dark Arts," muttered Professor Quirrell, as though he'd rather not think about it. "N-not that you n-need it, eh, P-P-Potter?" He laughed nervously. "You'll be g-getting all your equipment, I suppose? I've g-got to p-pick up a new b-book on vampires, m-myself." He looked terrified at the very thought.
What sort of magic do you teach implies that you already teach it. And Quirrell does not mention that he is actually starting to teach a new subject that he has never taught before. Also, Hagrid does not appear surprised at all that Quirrell teaches Defense Against the Dark Arts (though it is possible that he would have been informed of the position change in advance).
Shortly thereafter:
Hagrid grinned at Harry.
"Told yeh, didn't I? Told yeh you was famous. Even Professor Quirrell was tremblin' ter meet yeh — mind you, he's usually tremblin'."
"Is he always that nervous?"
"Oh, yeah. Poor bloke. Brilliant mind. He was fine while he was studyin' outta books but then he took a year off ter get some firsthand experience.... They say he met vampires in the Black Forest, and there was a nasty bit o' trouble with a hag — never been the same since. Scared of the students, scared of his own subject now, where's me umbrella?"
Hagrid's information seems to be referring to something that happened a while ago, not something that just happened. Specifically, he says that Quirrell "has never been the same since". If this just happened what is the "since"? Then he says that Quirrell is "scared of the students". How would Hagrid know this if Quirrell hadn't yet taught since returning? Finally, Hagrid says that Quirrell is "scared of his own subject now". Again, how would Hagrid know this if Quirrell was only about to start a new subject?
Furthermore, the start-of-term feast provides further evidence that Quirrell had already been teaching Defense against the Dark Arts the year before. Unlike every other year when
Dumbledore announced new teachers, he made no announcement for Quirrell. This would be especially odd if Quirrell had not been teaching at all the year before, but even if he had been teaching the year before Dumbledore still probably would have announced the position change. After all, in Half-Blood Prince he announced Snape as the new Defense Against the Dark Arts teacher even though Snape was already a teacher:
"Professor Snape, meanwhile," said Dumbledore, raising voice so that it carried over all the muttering, "will be taking the position of Defense Against the Dark Arts teacher."
(Though perhaps one could argue that he only announced Snape because he had already announced that Slughorn would be teaching Potions.)
Furthermore, at the feast Harry asks Percy about Quirrell:
"Who's that teacher talking to Professor Quirrell?" he asked Percy.
"Oh, you know Quirrell already, do you? No wonder he's looking so nervous, that's Professor Snape. He teaches Potions, but he doesn't want to — everyone knows he's after Quirrell's job. Knows an awful lot about the Dark Arts, Snape."
The fact that Percy calls it "Quirrell's job" rather than "Defense Against the Dark Arts" and the fact that he even knows that it's Quirrell's job indicates that Quirrell had already been teaching Defense Against the Dark Arts previously.
We know that JK Rowling has claimed that Quirrell was previously teaching Muggle Studies, but as per the above that is quite a stretch. Thus, if Quirrell has lasted for more than a year there must not be a curse on the job.
Further evidence that there is no curse comes from a passage in Chapter Nine of Order of the Phoenix:
”What d’you mean?” Harry asked, jumping down beside them.
“Well, we overheard Mum and Dad talking on the Extendable Ears
a few weeks back,” Fred told Harry, “and from what they were saying, Dumbledore was having real trouble finding anyone to do the job this year.”
”Not surprising, is it, when you look at what’s happened to the last four?” said George.
”One sacked, one dead, one’s memory removed, and one locked in a trunk for nine months,” said Harry, counting them off on his fingers. “Yeah, I see what you mean.”
This seems to clearly imply that prior to Quirrel there were no strange circumstances surrounding the job. If there truly was a curse, we would expect George to say “look at the last thirty” rather than “look at the last four”.
There is also a comment from Hagrid in Chapter Seven of Chamber of Secrets that calls the popular narrative of the curse into question:
"He was the on'y man for the job," said Hagrid, offering them a plate of treacle fudge, while Ron coughed squelchily into his basin. "An' I mean the on'y one. Gettin' very difficult ter find anyone fer the Dark Arts job. People aren't too keen ter take it on, see. They're startin' ter think it's jinxed. No one's lasted long fer a while now.
This implies that it was only recently that people had even thought that there was a jinx at all, and it also implies that there have been people who lasted more than one year, though they did not last "long".
Moreover, even when Dumbledore notes that they haven't been able to keep a teacher for more than a year since then, he makes no mention of a curse. Even if we assume that the correlative factor is Voldemort not getting the job, there is no indication that the direct cause was a curse. There could be any number of other explanations. Maybe someone was Imperiusing teachers to leave after one year. Maybe someone was deliberately trying to kill off teachers or incapacitate them in some way. This could have been related to Voldemort being denied the job, even if it wasn't via a curse. For all we know, someone might have wanted people to think that there was a curse even though there wasn't.
True, some of these theories seem unlikely, but the point is that in the absence of a controlled study there is no way to truly determine what the cause was. Additionally, it is possible that the "curse" is self-fulfilling. After the first few years of teachers not lasting (which could have been a coincidence) other candidates might have gotten scared off because they thought that there was a curse. With all the good teachers hesitant to take the job, by default new teachers will likely be the type that wouldn't last in the first place. And some of them might further have chosen to leave after a year thinking that it would exempt them from a worse fate if they tried to stay.
Perhaps most importantly, what in the world does it even mean for a job to be cursed? We don't really ever find an equivalent type of magic. If you could simply put a curse on a non-tangible entity, which can affect (and even cause death to) individuals not in your presence, the entire nature of the Wizarding world would be different. Instead of battles, wars, and duels, anyone could simply curse the job that their opponent holds. Surely, Voldemort would have cursed the position of headmaster and someone (probably not Dumbledore because he would be too noble to do it) would curse the position of Chief Death Eater, among many other possibilities.
In short, there does not seem to be any real evidence that there was a curse, nor does there seem to be any real evidence that such a curse is even a possibility.
