In answer to the question, you clean a lion's teeth very carefully and with the lion completely sedated. I think they used more than "happy gas" on him as you can see.
Since we have a zoo pass for the Metroparks Zoo, we often go just to get out of the house and get some exercise walking around. One of our stops for a peek in is always the Sarah Allison Stefee Center for Zoological Medicine. The last time we were fortunate enough to catch the veterinarians at work, we watched a root canal on a Mandrill monkey. This day, our attention was directed to the procedure they were doing on the male lion of the zoo.
The Center is a hands on learning place for children who can don a lab coat and pretend to be a vet while learning about the animals and their diets and health. There are several viewing areas where you can watch them work on some of the zoo animals when it's appropriate.
The lion was definitely out of it but I had to laugh when one of the guys got his arm tangled in the IV tubing and almost pulled out the IV. That could have been a rude awakening not only for the lion but the team of eight or so working in there on him as I'm sure that's what was keeping him sleeping peacefully.
We weren't sure what all they were doing. One zoo volunteer said they were just giving him a checkup. They were definitely cleaning his teeth though and we were able to watch the close up action on a video screen. I don't want to know what they were doing at the other end with his leg up in the air. Do male lions get a prostate exam?