There could obviously be psychiatric issues in dentistry, such as phobias. A good dentist could be quite therapeutic in this regard.
But there are a other dental issues that have to do with psychiatry.
For example, having unhealthy gums probably causes increased transient bacteremia each time one eats; this is caused by the mechanical stimulation of gums with a high bacterial load leading to leakage of bacteria into the blood circulation. This would not be expected to cause a systemic infection, but it would stimulate an immune response. Some research suggests that this type of recurrent phenomenon causes heightened systemic inflammation, which in turn stresses the brain.
Here are some references which show a relationship between gum disease and systemic inflammation; this causative relationship is further associated with increases in the risk of various systemic diseases, and overall mortality.
The existing research shows a link between oral disease and increased risk for various other diseases, such as cardiovascular disease. We can hypothesize that any factor increasing risk for cardiovascular disease would also be deleterious to the brain, as it would affect the brain's very sensitive vascular system. Not much research clearly proves this risk. Here is a reference which starts a discussion on the subject:
Here's another, suggesting that controlling or preventing gum disease is a preventable risk factor for Alzheimer's Disease:
So, it is a relevant part of preventative mental health care to take good care of your teeth, including regular dental visits!
There are other overlaps between psychiatry and dentistry. Many people, when depressed, neglect daily dental care. Psychotic symptoms can arise over dental issues (e.g. believing there is a transmitter implanted in a filling). Various overvalued ideas can persuade people to seek arguably unnecessary dental procedures (e.g. regarding mercury amalgam filling removals). While mercury is likely to be of some risk, e.g. regarding the development of autoimmune reactions, it is likely that many people overestimate the degree of risk, or falsely attribute symptoms to the type of dental fillings they have. Therefore, a business may arise of expensive filling replacements which are medically unnecessary. Here are a few articles about this: http://www.ncbi.nlm.nih.gov/pubmed/16042501
Another dentistry/psychiatry intersection has to do with cosmetic dentistry, and orthodontics. Orthodontic treatment is associated with at least a temporary increase in self-esteem, and possibly even an improvement in motivation. (reference: Karen Korabik, "Self Concept Changes during Orthodontic Treatment", Journal of Applied Social Psychology, 1994, 24, 11, pp. 1022-1034). Korabik's earlier work showed that orthodontic treatment led to individuals being perceived in a more positive way by others, with regard to impressions of personality, intelligence, as well as appearance (even, for example, based on pictures in which teeth were not visible). (Basic and Applied Social Psychology 2, 59-66, 1981). This phenomenon illustrates a problem with human nature, that we would infer things about one another based on superficial factors. The power of such superficial phenomena, relationally, is usually also short-lived and superficial, yet I do think that simple esthetic practices, if available, could be at least a small positive in affecting interpersonal dynamics, and therefore be a small positive influence in maintaining a healthy mood.