Concerns with washer and dryer in walk in closet
All dryers need a vent. Gas and electric dryers need a vent to the outside of the house to allow all the moisture to escape from the dryer. 

All-in-one washer/dryer combo units don't need an air vent, but need a water drain to remove the water from the drying stage (which you need anyways to drain the wash cycle). 

The walls are irrelevant. So don't worry about those.

The bigger concern would be the floor. If this is a high efficiency front-loading washer, it will be prone to lots of lateral shaking. Most wood framed floors are ill equipped to handle the speed of front loader washers. It's usually recommended that they be placed on concrete slabs. 

As such if the closet wasn't designed for a washer/dryer, then you probably need to beef up the floor as much as you can to reduce the shaking.