When the size of the index array of arraymesh exceeds a certain value, will the storage type of the index array change?
No, it wraps. In any case, it's a 32-bit integer. Which means it can go up to 2,147,483,647. There is probably no way you can render a model with 2 billion polygons, so I don't think you will ever reach the limit.
https://docs.godotengine.org/en/stable/classes/class_poolintarray.html