If there were such a place as Shangri-La (Lost Horizon) where people did not age, would it follow that other mammals would also not age?

Yes, it was fiction. But think outside of the story.

In Shangri La, people lived for hundreds of years – aging very, very slowly.

Would it also follow that mammals, reptiles, insects—that other living things would also live much, much, longer? What about foods that are grown?

Could a 200 year old guy be eating a 60-year old tomato??