Anything you want:
Unfortunately, the devil is in the details. The nature of physiological response is extremely variable, and depends greatly on a very large number of factors. How did the limb get removed? Many types of injuries resulting in these injuries result in numerous other injuries. If your fairy was captured and mutilated on purpose, they likely suffered other traumatic injuries.
What is the vasculation of the area damaged? What do the muscles involved attach to? A dragonfly wing is unlikely to have extensive vasculation, and slicing them off cleanly while leaving a stump would probably have minimal effects. Ripping them out of the sockets is likely to cause so much traumatic tissue damage that massive blood loss may occur, as multiple tissues are torn apart. The fairy may suffer cognitive deficiencies due to extended periods with inadequate blood supply to keep the brain alive.
Infection kills most people in these situations. The more traumatic the injury, the higher the likelihood of infection. Gangrene and necrotizing fasciitis can take a wound and make it deadly. This could be how your fairy lost it's wings, if an infection were not dealt with (but would be unlikely to affect both wings at once).
If the musculature of the fairy has the two wings working together, the loss of one wing could be problematic, as the resonances needed for flight wouldn't work but (for example) the remaining wing might spontaneously vibrate, causing excruciating pain as damaged muscles try responding.
Healing at the site may be problematic. In the case of the loss of a boney limb, bony growths may appear as the limb attempts to regrow while lacking the underlying extracellular substrate. Follow-up surgeries are often needed so the stump doesn't have jagged bony growth inside causing excruciating pain. What the fairy has is uncertain. Perhaps they spontaneously grow tiny, malformed wing material that must be removed, or they might have none of that. They may have a grotesque version of the wing grow back, making them appear deformed. It's up to you.
Something a lot of people don't thing about is phantom limb syndrome. The mind and body are still anticipating the presence of the lost limb, and people can experience sensations like pain or itching in a place where they have no limb.