I've had an infection in my gums ever since getting veneers on my front teeth, can I get them off?

I always preface by saying that xrays and photos are helpful.  First determine what exactly are the signs and symptoms, the origin of the infection and the diagnosis.  It may not be necessary to remove your veneer(s).  If you have an infection and determine that the nerve of the tooth is dead, it's possible to save the veneer and have a root canal done from the back of the tooth.  If by infection, you are referring to swelling, redness, inflammation at the gum line, it may be a poor-fitting veneer that is catching food/debris or the margin of the porcelain was placed excessively submerged under the gum.  In that you you should consider replacing the veneer.  Another possible scenario is there may be excess residual cement that wasn't completely removed that is causing the inflammation.  In that case by removing the cement can potentially help clear the inflammation.  Either way you may want to seek the expert opinion of an Endodontist, one who specialized in root canals and/or Periodontist (gum specialist).