Why do breasts feel like they are getting pulled when doing push ups?

The most common placement of breast implants is in the submuscular tissue plane.  Your description of the sensations you experience is typical for implants placed under the pectoralis major muscle.  At 3 months postop from your surgery, you remain in the early phase of healing.  Stabililty of the implant placement occurs 6-12 months after surgery.  With your implant sizes, you likely have a beautiful, fully developed chest and therefore, I would recommend avoidance of exercises designed to isolate and develop the pectoralis muscles.  Over time, the implants will stabilize and the sensations you are experiencing will subside.  If you enjoy weight lifting and cross training, you can still perform exercises for core muscle groups, bicep, tricep, thighs, legs, back(including latissmus) and shoulders.  You have very little to be gained by performing bench presses, pushups or pectoralis flies.  If you perform yoga, you can substitute other positions for the push up and planks.  Further advice can be obtained from exercise trainers or physiotherapists.

Breast implants are often placed submuscularly, that is under the pectoralis major muscle. This muscle extends from the medial chest wall to the humerus or upper arm bone. When the muscle is contracted, the implant may move or feel as though it is being pulled.