Good news, dental veneers are not known to be painful procedure because we're not working on the nerves of your teeth, unlike in a root canal procedure where absolute numbness is required. It's typically a pain-free procedure.
Porcelain veneer installation procedure vs Root canal treatment process
After your dentist numbs your teeth and gums, you will not be able to feel much of the procedure. The mild trimming of the enamel layer is usually painless. In some cases, when I employ minimally preparation techniques, I'm able to do it without any local anesthesia. However, this has to be assesed on a case by case basis.
I am sorry to hear that you have a phobia of dentists. Don't worry, you are not alone. Many of my patients have the same phobia too. If you're feeling nervous before the procedure, you can always opt to have some laughing gas or nitrous oxide to calm yourself down.
One more way to over come the situation is by I/V sedation. We usually have an anaesthetist on board when performing this. They take care of your nervousness and pain control while I work on your teeth.