In theory, using commas (or not) would be determined by if the information is nonrestrictive or restrictive.
But I can't imagine a situation where everyone would not necessarily include grandma. Otherwise, I'd expect it to read something like this:
He made everyone except grandma cry by telling his story.
That version has no commas because grandma is being isolated from (restricting) the larger group. Note, too, the different location of cry in that particular construction.
So, practically speaking (even if not in terms of theory), I can't see commas not being used in that sentence.
In short, I would use:
He made everyone cry, including grandma, by telling his life story.
On the other hand, I find both including grandma and by telling to be a bit unusual.
If I were writing the sentence, I would find the following (with commas) more natural:
He made everyone cry, even grandma, when he told his story.