Is it possible for someone to be allergic to dissolvable stitches? And if so, what would the reaction be?

This isn't as uncommon as you may think.  There are usually two types that promote a response in those who are sensitive to these sutures.  These two are usually Absorbable Poly(glycolide/L-lactide) Surgical Suture material and Absorbable Gut Suture material.

If you experience redness and itchiness and sometimes pus formation -- you very well may be allergic to the dissolvable type sutures. Sometimes the symptoms may not show up until about 3 to 4 weeks after surgery, if this happens, antibiotics can be given but it usually reoccurs.  The best thing to do in some cases is to remove any of the left over material and replace it with nylon sutures.  These sutures are usually removed in 10 days.  Another option can be tissue glue although this isn't very mainstream for rhinoplasty.

Also be advised that when the skin gets red and itchy around a suture, be it absorbable or non-dissolving -- usually it is time for them to be removed.  Your body knows and will tell you when your healing.  Besides healing skin always turns itchy.  Know the difference between typical irritation and an allergic reaction.