The rat in Ratatouille will probably die within two years from the end of the movie
Rats don't live long, the first result on Google says that they live 2-4 years on average and the longest living rat in captivity lived to be 7 years old.
I like to think that after the rat's death everyone will treasure the good memories and start another restaurant of their own. But it's more likely that they'll be depressed and miserable for the rest of their lives without being able to tell anyone their story.
Not all of them would be smart enough, but rat generations are crazy fast. Breed the smart ones with the other smart ones and you could be franchised nation wide in a couple decades.
Just a bunch of inbred genius rats yanking on some minimum wage teenagers head.
Hell, you could probably charge the human because an average person would learn how to cook pretty fast. It would be the best culinary school in the world.
My kid really loves the 2000s Alvin and the chipmunks movies. Dave is always talking about how they need to be like normal kids and they need to plan for the future, but on a lark I looked up how long a chipmunk lives and by the third movie they'd be like 50 in chipmunk years.
I think we'll just assume whatever Gene makes them talk makes them live really long. Let's do that.
Rats reproduce as if there was no tomorrow. Remy might be dead in two years, but his sons will run the place and a dozen other branches given the chance.
In a world where rats are intelligent beings capable of communication, who says the don't live like 10 or 20 years? It's fiction and, if not stated otherwise, you can believe whatever you want ;)