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

The breasts are being pulled because of the inadequate release of the muscles that is leading to animation deformity.  It is relatively easy to fix with a periareolar remove/replace with capsulectomy (scar removal) and muscle release.  

It is quite normal to have some movement of the implants with heavy pectoralis muscle contraction if your implants are under the muscle.  Usually by 3 months things have settled down enough to allow you to do chest exercises.  Perhaps you can build up to push-ups by starting with lighter exercises, like chest-flys using light weights.  I would also touch base with your PS about your concerns.

It is normal for implants to pull during chest exercise considering you did go under the muscle. As you are only 3 months post op it may take a bit longer for things to heal. You should adress your concerns with your plastic surgeon if it continues to bother you. 

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. 

Not to worry  It is quite normal to have discomfort and implant movement, especially when performing exercise in which you are isolating the contraction of the pectoralis muscle.  Over time while the implants will still move however the discomfort and tightness will abate.