Breathing Problems and GERD After Tummy Tuck

You did not give any details about your body habitus prior to surgery. It can happen in fairly large people with a protruding abdomen, that when they undergo an abdominoplasty and make the abdomen flat, the intra-abdominal contents are squeezed into a smaller space. This in turn pushes against the diaphragm, and thus limits expansion of the lower lung segments.
I lay my patients down in the supine position on my examination table and examine them that way as well. If their abdomen does not go flat, I tell them that they are not an ideal candidate for a tummy tuck. If I would tighten their abdominal wall to make it flat (sometimes this is not possible), they would end up with the symptoms you are experiencing. For this type of patient, I recommend weight loss with diet and excersise program prior to surgery or if they are morbidly obese, lap band or gastric bypass prior to an abdominoplasty.

This is possible and the most likely cause of your symptoms. This is not a cause for concern over the longterm.
However, there are potential causes of shorteness of breath which are an emergency. The most severe concern is a pulmonary embolism. You should relate these symptoms to your physician as soon as possible.

With significant tightening of the rectus muscles of the abdominal wall, there is a decrease in intra-abdominal space. Ones internal organs will be squeezed and this could increase pressure on the diaphragm. I have heard this complaint in the past but rest assured it is a temporary condition. The pressure should gradually diminish and your symptoms should subside.