Q:

Warlock character in Phandelver is OP'd?

One of my players has made a Warlock.  He can cast Shield from the Staff of Defense. With Mage Armor and Shield up he's got 22 AC... at level 3. 
We are playing through the Mines of Phandelver and nothing is able to hit him. He's owning everything hard. 
How I can go about getting through that defense? 

I had the party (all lvl 3's) get ambushed by one of the doppelgangers buffed with an Evil Mage spell list along with 4 Bug Bears and they couldn't make a dent in him.  She cast magic missile and could not hit due to shield.

Ultimately I am curious: how I can adjust the difficulty more appropriately so that these guys are challenged enough?  Adding more monsters doesn't seem to be working they are mowing everything down fast.

A:

I suggest you carefully read through the magic items and spells your players have so that you understand exactly how they work and what they do in order to avoid this kind of confusion.
The Staff of Defense says:

With the staff in hand, you can use your action to cast one of the following spells from the staff if the spell is on your class's spell list...

Shield and Mage Armour are not on the Warlock's spell list so they are unable to cast them from the Staff of Defense.

A:

Shield cannot be cast as a bonus action, it's a reaction  spell. However, it isn't a warlock spell so a warlock can't cast it, not even from a magic item. Neither is Mage Armour but this is less problematical as a friendly sorcerer or wizard can't cast it on the warlock or he could have it through Armour of Shadows.
If you let players do things the rules actually don't allow then prima facie they will be overpowered.
However, let me assume that you meant wizard or sorcerer instead of warlock because that could be a thing.
Shield lasts 1 round and can be cast a maximum of 7 times a day (6 spell slots + 1 through meta magic or arcane recovery). If the PC is using all their spell slots for this then they are not using them for other things. This tactic is inherently self-limiting.
Notwithstanding, there is a really easy way to overcome this. The bugbears grapple him; their Strength (Athletics) is +2, his Dexterity (Agility) is +4/+6 depending on proficiency (he must have 18-19 Dexterity to get the AC of 22). This gives each bugbear a 42.75/34% chance of success; not great but better than the 5% they otherwise have (but if they do hit its a critical). Once they succeed, they push him prone (same chances) and then wail on him with advantage (9.75% chance per round per bugbear of a critical). In 7 rounds he's used all his shields and in 9-10 he's dead.
Oh, and a well played doppelgänger always kills people when they are asleep! Ambush predators do not engage in fair fights.

