medication can cause symptoms like this. I took some anti-depressants once for the treatment of fibromyalgia and I had similar symptoms (palpitations, chest tightness) and they resolved once I went off the medication. Take your pulse when you notice it and tell your doctor you want this investigated.
I don't understand what "compression" of the chest is or means. As for a fast heart beat, I ask "how fast"? If it is under 100 beats per minute at rest it may not be a problem. A HR below 80 would be better, I think.
Some medications can cause the heart rate to increase, but if it causes the HR to be too high, say over 100 at rest, a change in medication dose or type may be in order.