They could, wild animals do get dental issues. The differences in their diets and how they are built to eat those diets can make some differences (i.e. Wolves and dogs have more differences than most people think, down to teeth) in how long build up and things may take as far as gum infection goes. As to how it would be resolved, it wouldn't be usually. If it did happen then without the medical care that is provided in captivity and domestic animals they would suffer the consequences and effects and could eventually die from it, not always in a nice way either. From gingivitis or a tooth abscess they may be unable to eat, go septic or heart, liver or kidney or other issues that result without medical care and suffer from those until death. Most animals live longer in captivity than they do in the wild, some significantly and one of the huge factors is generally monitoring and the medical care they receive in a captive situation that they don't in the wild. Here, we can properly diagnose and treat, out there it goes unnoticed and untreated shortening their life span and may result in a lot more suffering than a captive situation where if we can't help, we humanely end their suffering. The medical care results in more longevity as well as quality of life. One example that is an extreme is the African Serval ( a type of wild cat) which has also made it's way into the exotic pet trade (crazy little things). Average life span in the wild? About 2 to 4 years due to predators and lack of medical care ect. In captivity with decent care? 12 to 14 years. Quite the difference there just from going from 'wild' to captive and the differences that makes (not suggested as a pet and definitely not the average person, lol). Animals in zoos and sanctuary's and other situation do have veterinarians that specialize more in those types of animals or specific ones and regular medical care (and monitoring) is typically performed though not always in the way they are with domestic pets (may have to be sedated to perform a thorough exam, blood testing, ect.). A wild animal in a captive situation would be given appropriate antibiotics and they have dental work done but in the wild, they suffer the consequences of not being diagnosed and treated whatever the problem is.
