Stomach sleeping has fewer repercussions than side sleeping, however, it too can cause tightness and some muscle issues. Most notably, the head has to be turned to one side or the other. Even when you alternate which side you face, your neck and shoulders will still accumulate tension from being turned to one side all night long. This can cause neck and shoulder pain, tightness in the upper trapezius and sternocliedomastoid, sinus draining issues, and headaches.
Similarly to side sleeping, pressure on the side of the face and jaw can cause tension for the temporal mandibular joint (TMJ). This can cause popping of the TMJ or grinding / clenching of the teeth. This can also cause tension headaches that center around the temples.
Depending on body position, stomach sleeping can cause tension in the hips and lower back. This tension is due to an outward rotation of the hip on the favored side the head is turned. The body compensates this way mainly because when sleeping on your stomach, your feet are pointed directly into the mattress. In order to prevent this uncomfortable position, the dominant hip rotates outward, creating tightness in the glute muscles and puts pressure on the sacroiliac joint. The non-dominant hip can sometimes rotate slightly inwardly, but usually also causes the body to point the toes, creating contracted tibialis posterior, gastrocnemius, and soleus muscles.