According to Guide to a Healthy Cat (Elaine Wexler-Mitchell, 2003) male cats that have just been neutered will still be able to easily get erections for a few weeks after their operation. This is because it takes one month for testosterone to completely leave the cat's body. However, the cat still will be able to get an erection because all that is needed is a rush of blood to the penis. The penis will also still be capable of sensation for the cat. Removing testicles will not affect that. However, removing testicles does affect sex drive, so a neutered male cat is not usually interested in sex unless a female cat in heat walks directly in front of him. Cats that have been neutered after experiencing sex will get erections and try to masturbate on stuffed animals or soft objects after neutering, although Cat Owner's Home Veterinary Handbook (Delbert G. Carlson, DVM, et al; 1995) claims this rarely happens.
Frequent erections without any additional sexual behavior like pelvic thrusting means that another health problem is occurring in the neutered male cat that gets erections. If he is constantly licking himself and seems to be urinating outside of the litter pan, he could have a urinary tract infection. The infection will cause the penis to descend and sometimes become erect. According to cat health and behavior advice website Paws and Effect, constant erections in neutered male cats could be a sign of a neurological problem such as a brain damage. The brain sends out chemical messages to move certain body parts, such as the penis.
Another serious medical condition is priapism where the male cat's erection will not relax. This can potentially kill the cat because the cat could have a blood clot that is keeping the blood flow in the penis from going back into the body. According to the study "Priapism in Seven Cats" by D. A. Gunn-Moore, et al; Journal of Small Animal Practice, 28 June 2008, three of the seven cats studied with priapism were neutered and those three had blood clots (or thrombosis). Another complication from priapism could be feline infectious peritonitis, a problem with the cat's white blood cells that develop a virus for an unknown reason. The cat's immune system in the blood attacks the cat as it's developing a potential lethal viral infection. Feline infectious peritonitis usually happens after the cat had a problem with blood circulation, such as priapism.