After his decades of loyalty to the Corleone Family, wouldn't asking Tessio to betray the family be risky and start another war if Tessio reported to Vito & Michael?

Tessio may have approached Barzini.

Vito saw that move coming from a mile away. He even told Michael specifically how it was going to go down. So that raises the question of who Vito was really concerned about coming after Michael. Was he preparing him for Barzini's next move or did he foresee one of his two top men betraying the family.

Maybe Vito saw it from a mile away, but he wasn't a psychic. He couldn't foresee every move. It was clear, it had to be someone close to Michael - Tessio, Clemenza, Rocco, Al Neri, Carlo, Tom, Fredo. Their were advantages and disadvantages by approaching each of these men. I'm not sure whether it was Tessio who approached Barzini. Barzini wouldn't wait too long, he wanted to get rid of Michael as soon as possible.
Still, it was dangerous to approach Tessio, just as dangerous as it was for Brazi to approach Bruno Tattaglia.
Everyone of the guys mentioned above - except perhaps for Neri and Rocco - had their reason to be unhappy with Michael.
- Neri was simply too close to Michael.
- Rocco was the security guy (as I understand it).
- Approaching Carlo would be the most stupid move. Everybody would have smelled the rat immediately.
- Fredo had been humiliated by Michael and by his father, but maybe too far away. Roth later realized that approaching Fredo was the smartest move one could do.
- Tom had been dismissed as consigliere. I wonder if Barzini could have considered him.
- But Tessio and Clemenza suffered immediateley from the "Keep your hands still"-tactics of Michael. So it had to be one of them.

Vito foresaw the betrayal because of his cunning and because he had been doing "business" for so long. The Corleones were seen as weak, fleeing New york, and presumably falling apart at the seams. Everyone underestimated Michael...well almost.
Going through the list:
Neri -- No way. He was Michael's man. It was Michael who brought him in and brought him close. It has been said Neri was Michael's Lucca Brasi. He had gone from nowhere to being at Michael's side. Also unlike most of the others he did not underestimate Michael.
Rocco - Made his bones by killing Paulie, and with Michael ascending, Rocco was broen off from Clemenza to form his own secret regime. Barzini could not touch him.
Carlo had already betrayed the family to Barzini. Once that was done Barzini had little use for Carlo. He was a witless traitor.
Fredo - Still more or less loyal, and pretty much out of things in Vegas, there was no way Fredo would have the credibility to tell Michael he was in a position to make a deal and guarantee Michael's safety.
Tom - Tom was the ultimate insider, even when he was "out." He intuited there would be a betrayal, and he intuited there was a secret regime. He did not underestimate Michael.
Tessio and Clemenza - The most likely suspects. They had muscle, they had influence and they could be approached by Barzini who promised them they could keep their territories, run their family and not go down in flames with the collapsing Corleone family, if only they would betray Michael. In the end Clemenza remained loyal, and Tessio approached Barzini. We know this because at the funeral when Tom guessed wrong and thought it was Clemenza, Michael said that Tessio was always the smarter of the two, and it was the "smart move." This tells us Tessio went to Barzini.

How much time has passed since the funeral to the massacre?

I'm with Oli: Tessio approached Barzini. In the fishtank scene, it's Tessio who's complaining about Barzini. And, when Vito asks, "Do you trust my judgment?", Clem answers fervently, "Always, Godfather." Tessio hisses, "Yes-s-s-s-..." When leaving, Tess gives the left-hand handshake--a sure sign of perfidy-to-come.

Since Tessio brought Barzini's request for a meeting at Vito's burial, it seems likely that no more than a few weeks elapsed until the massacre.

I think the problem here is that we interpret the story from the end. We know already who the traitor is. So every hint to him appears large. All the hints to others appear small.
(For example, if FFC had given the story in GF2 another turn, say Fredo dies in Cuba, I bet that every-one here would say, that Michael could never ever kill his own brother.)

We never get to know why Tessio is supposed to be the smarter one. Obviously, he wasn't smart enough to stay alive. I doubt that Clemenza really trusted Vito's judgment at the point in the fish tank scene. Clemenza is convinced: "Mikey, you're wrong!"

Why couldn't touch Barzini Rocco, when it was easy to get in touch with Tessio?

Tessio said that.

If Tessio approached Barzini, I'm surprised Barzini went along with it. Wouldn't he smell a trap to see if they were setting Barzini up? Tessio was one of the founding members of The Corleone Family.

Tessio must of been very convincing to Barzini to hop sides with him. I wonder what he said if he actually approached Barzini.

You raise a very interesting point, Trilogy. Barzini's first reaction almost certainly would have been that Tessio was trying to trap him. Tess must have told him a long story about Michael's alleged failures and weaknesses. But, I'm guessing, Tess may have offered an alliance with the Corleone family after he took over--perhaps even to share the family's police/political power and some of their rackets. That would have made Tess, in effect, a subordinate of Barzini.

I think if the murder of Michael (and no doubt Clemenza) had come off, Tessio's days would have been numbered. Barzini would have no further use from him, and nothing is as untrustworthy as a caporegime who betrayed his Don to another Don.

... which raises the question whether he really was the smarter one. I mean, he knew the game. Traitors rarely survive, do they?

An alternative move could have been that Clemenza and Tessio form an alliance and kill Michael, Tom and Neri. Then make a deal with Barzini. Risky but not impossible.

As you know from the novel, Vito seemed to have anticipated the possibility of a plot against him by Tess and Clem, which was one of the reasons he kept them apart. But, a dual Donship (Tess and Clem ruling the Corleone family after disposing of the Corleones) would have been an impossibility.

Yes, Vito let them live in separate areas, but in a crisis situation nobody would be able to "keep them apart". I didn't mean a dual Donship, but something like Tessio as don, and Clemenza his underboss. Or Clemenza the don, Tessio the consigliere.

Vito played his last moments so beautifully. Keeping the inner circle so tight to flush out any last minute traitors or members who failed to stay loyal during the Corleone's appearance of their "weakest times".

Tessio was probably expecting a large pay out and disappear if his intel was provided correctly and the hit on Micheal was successful. Clemenza would eventually be a target too. Turnbull is correct about Barzini keeping a traitor around from Don to another which would make Tessio disappear fast.

I think Puzo just wanted it in storyline. Reason Tessio went against Michael is Michael's refusal to allow him or Clemenza to start own gang, he turns both down.
If I remember correctly, that happens after Vito dies. So Vito would have no way of knowing that go of them would go agisnt Michael. Michael turning him down seems to be only reason for him going to Barzini.

Because with Don Vito gone, he probably saw what he thought was an opportunity to become the top guy and he knew that of the two, Tessio and Clemenza, Tessio was the smarter, had the better head for business.

Completely unrelated to the topic but interesting none the less. There's a TV program called, "Inside the Actors Studio," and the host interviews one person for the entire hour. Several years ago he had Pacino on and they were talking abut his early life, that his father wasn't around, etc. The host, James Lipton, asks Pacino besides his mother, who raised him and Pacino says his grandfather. Lipton then asks, "And where was your grandfather from?" Pacino replies, "Corleone, Sicily." True story.