Mark this as a duplicate of MCPE-162063.

The items do indeed wear down, they simply don't display their wear levels anymore. Sorry for the confusion.

When creating a new world, all items (tools, weapons, armor) which have health values always remain at full health when used. Furthermore, when loading existing worlds, all items immediately get repaired to full health and no longer wear.

Steps to Reproduce:
Create a new world in survival mode, craft a wood pickaxe and use it on some stone or attack mobs with it.
Or, load an existing survival mode world and note the health of all items.

Expected Results:
The wood pickaxe should wear down, and all items in existing worlds should maintain their existing health levels.

Observed Results:
Items never wear down, whether they are used as tools or used as weapons. Armor takes no wear. And existing worlds load with all the player's items restored to full health - even if they aren't in the player's inventory and are in chests.