動く and 動かす: movement and machine operation
Compare bodies and objects moving with machines operating, and distinguish 動かない from 動かせない.
Ask someone to stay still
写真を撮るので、動かないでください。 asks the person to stay still while a photo is taken. The person addressed is the subject of 動く, read うごく.
机を少し右に動かしてください。 asks someone to move the desk slightly to the right. 動かす, read うごかす, marks the object with を and the direction with 右に.
Whether a machine works
新しい電池を入れたら、おもちゃが動きました。 says that the toy worked after new batteries were inserted. 動く can describe functioning even without clear movement to another place.
この装置を動かすには、電源が必要です。 means that the device needs power to run. 動かす can mean operating something, not only moving it physically.
Negation versus inability
この棚は重くて動かせません。 means that the shelf is too heavy to move. 動かせません is the negative of 動かせる, the potential form of 動かす.
棚が動きません。 simply says that the shelf does not move. The surrounding explanation is needed to know who tried, how hard, or whether moving it is beyond their ability.
Which expresses inability to move a heavy shelf: 動かしません or 動かせません?
Show the answer and why
Use 動かせません. 動かしません negates the action of moving it; 動かせません negates the ability to do so.