In some cases individuals will be allergic to certain types of (usually tropical) wood. If this occurs, there will be some swelling, redness, irritation and the skin may try and retract away from the jewelry. There's nothing you can generally do to "improve" this, so if this happens to you, take out the jewelry and try a different kind of wood or metal. (note: desensitization may happen, but as it is simple to swap materials, many feel it's not worth the risk.)
Wood allergies can also severely affect people manufacturing wood jewelry, since they may breathe in the dust which may be carcinogenic, and often causes allergic reactions. If you notice swelling or stiffness in your hands, perhaps associated with a "heavy feeling" in your arms, severe headaches and visual disturbances, a rapid pulse, dysphagia and diarrhea after working with a certain kind of wood, get away from that wood!
Drink large amounts of fluid and you should be fine in time. See a doctor if symptoms persist, and know that fatalities are extremely rare with this type of allergy.