Clearing out my inbox before I take a week off to go to Dragon*Con in Atlanta. Here's an email from Teng.
Rodents have rather interesting teeth. They continually chew on hard and abrasive materials, which would wear down the teeth of most other mammals. To counter this constant wear-and-tear rodents have open roots and their teeth grow continually throughout their lifetime. Normally this isn't a problem as the upper and lower teeth will rub against each other and the animal will chew on things that will keep the teeth worn down.
Problems can and do occur. Often this happens after a tooth breaks off, resulting in the opposite tooth not having anything to grind against. Misaligned teeth or jaws can result in the same problem, allowing teeth to grow longer than is normal and natural. It is also possible to have abnormal wear due to chewing habits, resulting in points, elongated teeth, and so on. Regardless of the cause, overgrown teeth are potentially a serious problem. Molars can rub against or even trap the tongue, causing pain and an inability to eat. Abscesses can result due to infection in a broken tooth or an overgrown tooth pushing into the gums.
It is impossible to stop a rodent's tooth from growing, regardless of whether it is a hamster, rat, prairie dog, guinea pig, or rabbit (and before I get comments, yes, I am fully aware that technically rabbits are lagomorphs and not rodents). When you have overgrown teeth you have very limited options. Initially the best option is to keep the teeth trimmed, which can be done by any vet with proper skills in these species. For the incisors you may be able to trim the teeth while they are awake, depending on the animal's behavior. For molars you always have to fully sedate them because it's nearly impossible to get that far back in the mouth while they are awake.
Other than trimming the other solution is to have the affected teeth extracted. This is much more difficult than removing them in a dog or a cat due to the tooth structure, and so it should only be attempted by vets who know what they are doing. You also have a problem in that you'll often have to remove teeth on the opposite jaw to prevent them from overgrowing. And you can't remove all of their teeth since they need to be able to chew and grind their food for proper digestion.
Yes, this can become burdensome. Yesterday I saw a guinea pig that is a regular patient of mine. The owner got her from Craigslist, and on her first examination I discovered that she was completely missing her lower incisors, the upper ones were growing back into her mouth, and her tongue was split and had healed. I suspect that she had some sort of injury that caused the lower teeth to become damaged and fall out, cutting the tongue at the same time. We see her every six to eight weeks to trim the upper teeth and keep them at a proper length. I have another patient, a rabbit, who has a misaligned jaw which causes her teeth not to properly contact each other. I have to trim her incisors at about the same intervals. Both owners know why and how this has to be done and have accepted that responsibility.
Teng, I wish I had better answer for you. If this is as serious as it sounds you may need to contact an exotics specialist, as many general practitioner vets don't have the experience or equipment to treat these cases. Talk to your vet about the various options and get their opinion.