Yes, lions do kill and eat cheetahs but this isn’t the full story. Two separate sources tell me that lions kill a lot of cheetah cubs.
Cheetah cubs have a relatively high mortality rate in the first few months of life. Two weeks after emerging from their den at eight weeks of age only 51% of cheetah cubs are alive. By the time they are four months old 27.8% of cheetahs which emerge from the den are alive. One study found that two thirds of cubs from 12 litters that died at the den of known causes were killed by predators. Lions were responsible for more than 80% of cheetah cub deaths by predators at dens. In those cases where the entire litter of cheetah cubs perished, the lion killed them all.
In a slightly bizarre twist to the story, lions rarely eat the cubs that they kill. This means that they do sometimes eat them which answers the question in the title but please note it is rare and in a further twist to the story, the cubs’ mother often ate their kittens’ carcasses after the lions had left the scene. Perhaps lions kill cheetah cubs to remove competition.
Another source on the Science Mag website confirms that lions rarely ate the cubs that they killed. In Tanzania’s Serengeti National Park 57% of cheetah cubs were killed by lions.
As for adult cheetahs, I don’t have information about there loss to lion attacks because the books that I have do not mention it, which strongly indicates that lions very rarely kill and eat adult cheetahs. It may happen occasionally for various reasons and under certain circumstances but I have no documented records of it. It must be very rare.
On the upside to what is a devastating story of cub mortality at the hands of this top African predator (other than the human) the mother who has lost her cubs becomes fertile within two weeks of losing them. A new family is quickly on the way.
Lions discover cheetah dens by noticing their mother sitting up with her cubs or by seeing a cheetah mother resting near her den. The lions actively search for cubs after spotting their mother. They kill them by biting them through the skull or spine. It sounds pretty gruesome but no doubt cheetahs do their utmost to avoid lions but one source says that the make their dens in lion territory perhaps because this is where there are more prey animals for both of them.
Main source other than stated: MK Laurenson High juvenile mortality in cheetahs and its consequences for maternal care. 1994 J. Zool. AND MK Laurenson, N Wielebnowski, TM Caro Extrinsic factors and juvenile mortality in cheetahs 1995 Conserv. Biol.