1) you have become sensitized to the lidocaine.
--caine type drugs do cause sensitization, in infants and children and those with sensitive skin

2) you may be reacting to the adhesive. If this is the case,
try to get any residue off your skin, when you remove the patch.
You may also be trapping chemicals under the patch, like soap residue etc.

People typically sweat under the patches, so any water soluble soaps trapped there can burn the skin.

I typically don't get residue from my Lidocaine patches. But the Salonpas do leave a residue and I sometimes get redness from them, but not often. People really complain that Lidoderms don't stick well. I haven't had a recent fill on mine, but it is possible they changed the adhesive. If this is the case, that would be one explanation. Contacting the company would be best to see if this is true.

Lidocaine allergy can be serious, so you need to contact
your doctor, and show him/her the rash.
If it only happens on one area, and not others, then lidocaine
allergy is not as likely as some other explanation.
Over the counter hydrocortisone cream should help take it
down.

Do not continue with them if the rash persists..the allergy can become serious
and systemic. It is important to determine whether you are allergic...before
your next dental appointment.