No, Messi will not leave. But this tiff is bad for Barca, there will be no winners.
If Messi is just constantly agitating about his contract, this is very destructive to the club, and there is absolutely nothing the club can do to fix it. They can't not give him whatever he asks for, and it's also just a terrible idea to let him re-write his contract every few months. Of course, this is Messi's club. Not the socios' club, not the board's club, not Rosell's club. Messi will get whatever he wants. He can name the next president if he chooses to do so, the socio's will vote as he tells them to. I would if I was a socio.
Barca stability has been a big edge compared to almost all other clubs. Laporta burned vast sums of money wastefully, and his accounting was not truthful. Rosell is a real businessman, even if you hate him you have to admit that his financial management is excellent. On the football side, both ran the club in a very stable way by the standards of this sport.
Messi by himself can put an end to this stability. He does seem to be poorly advised, he should just live with his recently renegotiated contract. The Catalan tax increases and the collapse of his tax sheltering have reduced his net income quite a bit. Of course, he is underpaid, team sports always underpay their biggest stars. Messi is worth at least 100M euros a year to Barca, 20% of the gross.
Faus may have been ordered to sacrifice himself in order so that the board/management could deliver a message to Messi's camp.