Why is Christian Bale not in the upcoming movie Batman vs Superman?

TL;DR:

Christian Bale isn't in the movie as he wanted to leave the role after the third Dark Knight film. Ben Affleck was the choice to replace him.

Long Answer:

Christian Bale** certainly said he was jealous, but that by no means indicates he wanted the role himself. To quote from USA Today:

The former Batman ended his tenure as the masked vigilante with The Dark Knight Rises, but that didn’t stop him from entertaining the idea of going back to the role, he told Empire Magazine (via Comic Book Movie). So even though he knew it was coming, he was still shocked when he heard Ben Affleck had been cast as Batman in Zack Snyder’s upcoming Batman vs. Superman: Dawn of Justice.

I’ve got to admit initially, even though I felt that it was the right time to stop, there was always a bit of me going, ‘Oh go on… Let’s do another.’ So when I heard there was someone else doing it, there was a moment where I just stopped and stared into nothing for half an hour.

But Christian Bale got over his existential crisis and even offered a couple pointers to Ben Affleck.

But I’m 40. The fact that I’m jealous of someone else playing Batman…I think I should have gotten over it by now. I haven’t spoken with Ben, but I emailed him offering bits of advice that I learned the hard way. I would imagine he is doing everything he can to avoid anything that I did.

In other words, he stopped because he felt it was the right time to stop. It didn't mean he still didn't get a little jealous at hearing who had replaced him - but ultimately, the decision to stop was his. He left the role of his own volition.

As for why Ben Affleck was chosen to replace him. From Zack Snyder, the director of the film:

Ben provides an interesting counter-balance to Henry’s Superman. He has the acting chops to create a layered portrayal of a man who is older and wiser than Clark Kent and bears the scars of a seasoned crime fighter, but retain the charm that the world sees in billionaire Bruce Wayne. I can’t wait to work with him.

Ultimately that decision has divided fans and only time will tell how worthy a choice he is as the new Batman.

Edit:

To add some further info on why Christian Bale left, this is from ScreenRant:

“We were incredibly fortunate to get to make three. That’s enough. Let’s not get greedy…Chris always said he wanted to make it one film at a time. And we ended up sitting there looking at each other, saying ‘We’re about to make the third.’ We never really knew if we were going to get to be there, but if that was how it was going to be, this was where it should end as well.”

Later in the same article, he is asked about what he knows about the upcoming Batman movie. He responded:

“I have no information, no knowledge about anything. I’ve literally not had a conversation with a living soul. I understand that they may be making a Justice League movie, that’s it… It’s a torch that should be handed from one actor to another. So I enjoy looking forward to what somebody else will come up with.”

It's a different continuity. A reboot of batman. The batman in Batman V SuperMan has a similar past, but not exactly the same. As Zack Snyder said:

This is a total and opposite reality from the Chris Nolan movies. It’s another universe, so we couldn’t hire Christian Bale if we wanted to, because he doesn’t exist in our world. Maybe we could hire him to play another part. We did talk about that briefly. I just wanted to hire Christian to play another part to make that obvious. Christian could play, like, Alfred with age makeup. No! Of course not. But you know what I mean. Even people at the studio would say, ‘Who are you getting from the other movies?’ And I was like, ‘Hey, come on guys, let’s all understand, it’s a different world.’ In the Batman universe that Chris Nolan created, Superman would have a hard time existing. That that’s why we did a reboot on the universe, so we could allow these characters to exist together. We needed to do that to have Batman exist in this world.

This also opens the world for a new Joker, giving way to Leto's Joker.

source: http://filmink.com.au/2016/zack-snyder-in-depth-on-batman-v-superman/