is it possible to be allergic to dentures?

I know of a few people who are hyper-sensitive to acryllic. So, yes, it is possible.

Edited to add: the worst case of acryllic allergy I've ever seen was a dentist. She could not even handle dentures without breaking out everywhere. Even gloves and a mask did not protect her. She had been a chemistry teacher before becoming a dentist and had been exposed to many toxic chemicals. Her doctor thought this may have been the cause of her extreme sensitivity.

Some lower-quality partial dentures might have metal frameworks that contain reactive metals such as nickel, which can and often do cause an allergic reaction.

As far as the plastic material in dentures go, documented cases of allergic reaction to the acrylic plastic used in today's dentures do exist, but are very, very rare.

It could be possible to be allergic to whateve the dentures were made out of or a chemical in the dentures.

I am having an allergic reaction to my new flexible Valplast partial and I have been reading about a lot of other similar cases! As soon as I started wearing it, I had a burning sensation and pain all around my tongue and gums. I stopped wearing it right away and now, 4 days later I still have pain and blisters on my tongue and gums. I am very unhappy about this and will never put that back in my mouth! Something is very wrong with the materials being used for these partials!

I haven't seen anybody allergic to acrylic but I often find people allergic to denture pastes and cleansers. They often don't get the cleaners removed enough or they are allergic to the adhesives.