Why did Tyrion betray Varys?

That's right, he didn't have to betray Varys, especially since Varys was the one who helped him escape after he murdered Tywin. It seemed to me like he was trying to save his own skin(Dany would propably burn someone anyway since she was paranoid and saw enemies all around her), thinking that if he ratted Varys out, he would prove his own loyalty and regret for not keeping the secret about Jon's true identity.

His character was already a mess for a while now, he keeps turning from one of the smartest people of this show into a complete idiot. He also used to be one of the most likable characters and this betrayal makes him a pretty horrible person. Varys risked his own life to help him escape King's Landing, but Tyrion couldn't keep his mouth shut. And the worst part is that deep down he knew Varys was right about Dany and he still acts like an idiot, so all those people who were massacred in King's Landing have Tyrion to thank (among others).

If Varys had kept his actions to just sending out ravens to everybody, I think Tyrion would have let it go. It was because Varys was actually attempting to poison Dany that Tyrion had to choose a side.

Because Varys was doing everything he could to spread the word that Jon had a greater claim on the throne. If he did nothing he'd be party to Varys complicity so he was forced to choose. He still held out hope Dany was the good and just ruler he thought she was when he originally pledged himself to her cause.

Tyrion's mistake was telling Varys, just like Dany pointed out to him, he was influenced by Sansa. The last conversations he had with Varys, they stated their positions, so Varys was gonna do whatever he was gonna do, knowing Tyrion was gonna do his job and most likely turn him in. Maybe he hoped Tyrion would change his mind, but he wasn't counting on it. And the same for Tyrion. He hoped Varys would change his mind, but it was clear that wasn't happening.

By the time he went to Dany, she already knew something was up. By being straight with her, he probably saved his own life, at least for a day.

I wouldn't be surprised if Tyrion dies in the finale. Probably killed by Dany in some way.

Varys needed to die for adding drama to the show. Everything else is minor details.

Tyrion needed to save his own neck. He watched Varys approach Jon and knew by Jon's reaction that Varys told him to be king. Tyrion knew that Jon would tell Dany so Tyrion had to tell her first or he would've been killed with Varys.