What in adhesive tape and glue makes some people allergic to it?

Latex. Many adhesives contain latex. Allergic antibodies (ige) can be formed to latex. Testing with a blood test may identify these antibodies however; false negative reactions are fairly common. (testing may fail to identify this problem accurrately) . There have been reports of severe allergic reactions to latex and if you are latex allergic, report this to all healthcare providers, including dentists.
Irritant reactions. Studies looking at people with self stated adhesive (bandaid/tape) allergy point to the reaction being and irritant (mechanical) reaction rather than an immunologic one. However you can be allergic to latex, or anything you might apply with the bandage.