Why didn't Voldemort remove the Jinx on the DADA post when he had Quirinus Quirrell and Barty Crouch Jr?

In addition to the excellent current answers, I think an alternative answer can be gleaned from what Voldemort did once he achieved power: He abolished the position.

Instead of Defense Against the Dark Arts, "Voldy" Hogwarts taught straight up Dark Arts to students. Presumably, Voldemort thought that the change of subject would be enough to remove the curse, assuming he even cared about it; He's shown to have a cavalier attitude towards his followers.

there are a few possible answers and I will review them here:

Voldemort intended to come back to life and rule the wizarding world within a year. therefore having one of his death eaters work at Hogwarts was unnecessary after the year past.

Quirrell- Was Supposed to retrieve the stone within the year. Voldemort would have than return to full power and have his death eaters with him. However, that might take time and Voldemort would still need for the meantime a spy in Hogwarts- No worry, Quirrell taught muggle studies before DADA, and it would be quite reasonable for him to switch back to a subject he isn't afraid of.

I also won't be surprised if Voldemort would just leave Quirrell to die at the end of the year, since in Voldemort's opinion Quirrell is a fool and a puppet who needs to be possessed and has no use to him after he returns to power.

He left Quirrell to die; he shows just as little mercy to his followers as his enemies.

BCJ- was pretending to be Moody, who probably insisted to Dumbledore to teach at Hogwarts only for a year, and it would look suspicious if he changed his mind.

Yeah, I'm staying just the one year. Special favor to Dumbledore.... One year, and then back to my quiet retirement."

However, there might have been a plan to keep Moody (BCJ) at Hogwarts for another year, since he will prove as a useful spy. that plan was probably changed when Snape came to Voldemort at the end of GOF, and Voldemort kept Snape as a spy rather than BCJ, for 2 reasons:

A- It will be suspicious if the retired Moody who just told Dumbledore he'll work at Hogwarts for only one year as a special favor would change his mind and want to work there more. (There is also a big chance Dumbledore recruited Moody specifically for one year, since he knew about the DADA curse and would rather Moody would leave on his terms than Voldemort's)

B- It will be really suspicious if after 25 years of a different DADA teacher, Moody would break the streak. Dumbledore will surely investigate into it, exactly the thing Voldemort doesn't want.

"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."

There is also a chance Voldemort's DADA curse was permanent, though we can't prove that.

Also, in both Quirrell's and BCJ's case, Voldemort was in a very weakened state and didn't have the power to remove the curse.

In conclusion: Voldemort didn't remove the curse because it would be unnecessary, undoable, required power Voldemort didn't have at the time and raise suspicion.

He didn't need to and there was no need to do so.

He only wanted the DADA job so that he could influence/recruit more and more wizards for his cause (i.e, his future Death Eaters). But remember, he was rejected for the position not once but twice. This is why Tom Riddle placed a curse on the DADA position: mere jealousy and the missed opportunity to influence young minds into joining his forces.

After Tom became Lord Voldemort, he had no reason to become a teacher at Hogwarts anymore; he had his followers and also the power that he craved in the first place. It seems likely that he simply forgot about the curse that he put on the DADA position.

However, he was overthrown when the killing curse bounced off of Harry and hit him, leaving him body-less, like a mere ghost. Thus, the plan with Quirell came into picture. And when Quirrell failed, Barty Crouch was put in place (NB: read the points below for each character).

Until the end of Book-5 (Harry Potter and the Order of the Phoenix), Voldemort's return was a secret. The removal of the curse at this point would prove that He has returned which, Voldemort, at all cost wanted to avoid. Following the events of Book-6 (Harry Potter and the Half Blood Prince) and the death of Dumbledore, Voldemort had no need to get rid of the curse.

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Additional notes

1. Quirrell

The main objective of Quirrel getting the job of DADA was to retrieve the Philosopher's Stone. And since Voldemort was at the back of his head at all times (literally), the job would have been much more quicker. Voldemort's motive at this point of time was simple: to finally get a body of his own using the Stone.

"See what I have become?" the face said. "Mere shadow and vapor ... I have form only when I can share another's body... but there have always been those willing to let me into their hearts and minds.... Unicorn blood has strengthened me, these past weeks... you saw faithful Quirrell drinking it for me in the forest... and once I have the Elixir of Life, I will be able to create a body of my own ..
_{Harry Potter and The Philosopher's Stone Chapter 17: The Man with two faces}

2. Barty Crouch Jr.

When Voldemort's attempts in the Harry Potter and the Philosopher's Stone failed, he realized on thing: He could not touch Harry. Thus he found the opportunity to get his body back by using the ancient magic the revive himself using his Tom Riddle Sr's bones and Harry blood. Barty was there to only recover Harry and get him to the graveyard for Voldemort's return.

“And what did Lord Voldemort ask you to do?” said Dumbledore.

“He asked me whether I was ready to risk everything for him. I was ready. It was my dream, my greatest ambition, to serve him, to prove myself to him. He told me he needed to place a faithful ser- vant at Hogwarts. A servant who would guide Harry Potter through the Triwizard Tournament without appearing to do so. A servant who would watch over Harry Potter. Ensure he reached the Triwizard Cup. Turn the cup into a Portkey, which would take the first person to touch it to my master. But first —”
_{Harry Potter and The Goblet of Fire Chapter 35: Veritaserum}

Given that Dumbledore was unable to remove the jinx, it could not have been easy to remove.

At both times that his followers held it, he was not yet recovered, a feat that he did not pull off until the end of Goblet. His efforts were bent on staying alive and recovering, neither of which were easy. He would have needed a powerful reason to take off the jinx.

Furthermore, in both cases, his plans could be carried out within the year timeframe of the jinx. If he had other plans for Quirrel or Crouch after the year, there was really no point in making life easier for Hogwarts by removing it. (It wasn't always terrible for the teacher. Snape left the post because he became headmaster.) And if he decided to remove it after, he would still have had time.