The gums behind my 4 top front teeth are swollen, hurt to the touch, and I can feel my heart beat there too. Should I be concerned?

Get it checked. This is something you should see a dentist about ASAP.. It could be an infection, or due to trauma. In the meantime take ibuprophen for the pain, brush gently, and rinse with warm saltwater.
Yes. Any swelling may indicate an infection and should be checked by a dentist as soon as possible. Even if it's not an infection, the dentist can determine what the problem is and how best to prevent and\or treat it.