Yes you can replace them with new veneers if existing veneers are iatrogenic and causing the inflammation of the gums.
I Can Replace my Veneer in Front of my Teeth After Treat the Gum Inflammation but Replace with Another Veneer or Crown Better?
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Replacing an existing veneer
Once the reason for the gum inflammation has identified and resolved, the most conservative restoration should always be done. In this case it would be replacing the veneer with another veneer.
Replacing a Front Veneer With a Better Porcelain Veneer
You're doing things in the right order. Always want the gums to be healthy before placing veneers or crowns. The fact that you mention you are wanting a new veneer tells me that either your had some unsightly gum recession around your old veneer, or that you just don't like the way it looks and want something better?
Replacing an old veneer is very straight forward. Your dentist will numb you like before, then very gently he will drill off the old porcelain veneer and any residual bonding material. Then he takes an impression and makes you a temporary veneer. Your impression goes to the dental ceramist who then makes the new veneer and sends it back to your dentist, who will place the new veneer. From beginning to end, the process takes about two weeks.
If your veneers were made more than five years ago I can assure you that today we some incredible new techniques and materials that give us stronger, longer lasting veneers, and also the new veneer should look much more natural than your first one. My favorite material for porcelain veneers and all-ceramic crowns today is the EMax material. It's like a breath of fresh air compared with the older and weaker ceramics that used to be the norm.
Dr. Norm Huefner