The muscle that extends the forearm when doing push-ups is the

Biceps brachii

Explanation:

Biceps brachii is the skeletal muscle which is involved in supporting the movement of shoulder and elbow. This muscle is found on the front of the upper arm between the elbow and the shoulder. The functions of this muscle are listed here -

It turns the palm upwards It flexes the forearm while lifting an object or performing biceps curl It also assists in forward flexion of the shoulder joint which occurs during bringing the arm forwards and upward.