Third eyelids are most often removed due to tumor involvement (usually SCC). See youtube video
Eyelid masses (generally SCC) may be removed using H plasty. It is important to maintain or recreate the eyelid margin. This surgery pulls up skin from lower on the face to create the margin. Really cool and works in all species.
Equine periocular sarcoids are generally not removable (removing them would leave a huge gap and cause exposure keratitis. We treat these with imiquimod (safe near the eye), cisplatin beads and/or other chemotherapeutic injections. Combination therapy can be useful.
JS Harper. How to Surgically Remove the Third Eyelid in the Standing Horse. 2009 Vol. 55 AAEP PROCEEDINGS
B Welker et al. Excision of Neoplasms of the Bovine Lower Eyelid by H-Blepharoplasty. Veterinary Surgery, 20, 2, 133-139, 1991