Everyone in the anime 'passes on' after coming to peace with the life they used to have, except for Otonashi. Unless I'm really confused, Angel seemed to imply that the reason Otonashi couldn't pass on is because when Otanashi was alive, he was an organ donor who gave away his heart.
This explanation doesn't make sense to me. Does that mean that all organ donors are stuck in limbo like Otonashi?
Why couldn't Otonashi find peace/pass on to the next life?
Otonashi did pass on, although he was the last to pass on. He is assumed to have reincarnated at the end of episode 13, the final episode of Angel Beats, and it is heavily implied he meets the reincarnated version of Angel.
The reason why he couldn't pass on just yet was not fully explained, but we can try to form a conclusion from it, so it is not entirely speculative. First, he lost all of his memories and so he has to regain it to truly understand what he regretted. He did not end up in the afterlife normally like the other students had, so it's highly possible that he did not have a regret in the first place. It could be that he was in that afterlife in order to have Angel move on with her life, as she regrets not being able to give thanks to the person who gave her heart. Kanade, the Angel, was there before him and that normally wouldn't make sense since he died before her.
Another explanation is that he simply regrets not being able to achieve his dream of becoming a doctor or more specifically, to help people. He wanted to help people after his sister's death and because he died before he was able to reach his goal, it's understandable that he regrets not being able to see for himself that he saved a life. Once he sees that Kanade was able to live her life, that she was content with the life he gave her, he was able to move on shortly after.
I don't think all organ donors would necessarily have regrets. If they lived a full life, they would not be stuck in this afterlife school in the first place. Remember that you will only be in this afterlife if you have regrets. Otherwise, you would presumably just be reincarnated. Just because you donated your heart doesn't mean you get to be stuck in that limbo.
Sources:
Otonashi actually was not supposed to be in that world, but due to him losing his memory, he stumbled into that world accidentally as stated in the anime. But even after regaining his memory, he did not disappear. That's because although he did not have any regret in his previous life, he gained something that was necessary to be accomplished in this world - which was helping everybody to pass on. So in the anime, it doesn't show the exact moment of Otonashi passing on, but after the ending song, it shows reincarnated Kanade (Angel) with short hair humming to "My Song" and reincarnated Otonashi with his hat on walking pass her. Moments after he passed by Kanade, Kanade started to walk away, and Otonashi then turned around and extended his hand toward her back as if he was trying to tap her shoulder.
In Another Epilogue, it actually shows Otonashi not passing on. It's only like 2 ~ 3 minutes long and it shows a boy in the Battlefront world who finally figured out that something was wrong in the middle of a test. He caused a riot and the student council president walked into the classroom, and you can see that the student council president was Otonashi. Otonashi then told the boy what he should do (sort of, he kinda gave hints) and told him to come to the student council room if he had any questions. After Otonashi left the classroom, the NPCs murmured about how cool he was and if he had a girlfriend. Then one NPC said that there was a rumor that Otonashi was actually waiting for someone and you can kind of make a relation to the situation of the creator of Angel Player and Otonashi, who both found love in the Battlefront world although they were not supposed to.
In the ending scene of the final episode of Angel Beats, Otonashi is seen but reincarnated. This does specify that he did pass on. Otonashi was the final person to pass on and stayed for a little while before finally disappearing. In Another Epilogue, it states that he was waiting for someone, but that 'someone' already passed. More information on this includes that he was able to 'meet back up' with the SSS, but in a heaven or reincarnated.
Another thing is that he didn't specifically fulfill his regrets, but disappeared anyway. Some people can make up ideas for what really happened. We all know, based on the ending scene in episode 13, that he did pass on, but for reasons we don't fully understand.
The main reason why he may have disappeared is that he finally realized that he did all he could, and he did save Angel's life, gaining a 'thank you' from her at the end when she passes. He may have stayed behind long enough to realize he needed to leave.
Another way we know he passed on is that in the ending scenes when everyone is disappearing from the picture, in the end, we can see Otonashi disappearing too. Signalling he did pass on to either become reincarnated with everyone else or to go to heaven. The final ending isn't as sad as we all think for Angel beats, considering everyone passes to the afterlife, potentially meets up again, and they all live happy lives.
According to Angel Beats! Interview with Maeda Jun, Second Season??, the writer confirmed that he was able to move on, reincarnated, and meeting Kanade's reincarnation. He will live a happy life.
Maeda was asked what happened to Otonashi after the last scene that moved many.
His response:
It was no use being there by himself, so I believe he also left the world after that. Besides, he was rewarded. It wasn’t a bad life. It wouldn’t be like Otonashi for him to stay. He’s a forward-thinker…looking towards his next life.
In the 3 days [between EP12 and EP13] there were probably a lot of drama too, but in the end, all other members left for their next life after a satisfactory persuasion from Otonashi.
So, for the TL;DR people:
Otonashi left the world soon after. In the 3 days he also convinced other SSS members to leave the world as well.
You are confused on this matter. Kanade says that he would be able to stay at the afterlife school because he came there without regrets. His organ donation has nothing to do with this.
As far as we know, the afterlife school is designed to give people the chance to come over their regrets and when they reach a state of peace and happiness, they move on. Otonashi came without regrets and that's why the school can't help him make peace. That is why there will be nothing like Hinata's baseball match or Iwasama's "My Song" debut that makes him disappear. The only way for him to move on at this point is to find this state of peace and happiness on his own - but he has the choice not to seek it and stay helping people instead.
Eventually, he will regret that he has not moved on together with Kanade and later makes peace with this or he becomes the programmer or he just is president for some time and then decides to move on for some reason. The scenes where the characters meet in another reality may be the next life, or maybe they are just their own imagination. If they are real, then at some point Otonashi moves on and meets Kanade again.
There is a theory that Otonashi is actually the creator, so Another Epilogue could actually be a prologue and the ending was that shortly after Kanade leaves, Otonashi leaves as well as his final regret was not meeting her again. Shortly after he was reincarnated and hears reincarnated Kanade humming the song she created and Otonashi realists and taps her on the shoulder. The end.
As I would like to say, Otonashi did pass on, there is an evidence when he was about to go after the boy that can control people with his eyes. That means that he can pass on but there was just no scene of him passing on.
