Some people might have an allergic reaction to frangipani sap however most people will not feel any effects of getting sap on their skin.
You will usually not feel any discomfort even if you get frangipani sap in your mouth or eyes but you should wash it out before it hardens.
There is more of a danger with young children or animals who might choose to chew on a frangipani branch sucking the sap out for and extended period of time.
I have heard one story from a person whose dog was alone in the backyard. The dog chewed and ate quite a bit of frangipani wood but never got sick afterwards.