Although this is a broad question, due to the many various mental disorders that can be considered, it can be addressed by looking at common mental disorders that have been studied in LGBT research. Perhaps one of the most common mental disorders studied is depression.
The 2011 Illinois Youth Risk Behavior Survey provides information from Chicago high school students about depressive feelings (among other risk behaviors). It was found that students who identified as gay, lesbian, bisexual or were unsure of their sexuality were twice as likely to report depressive feelings than those who identified as heterosexual. In a study of midlife adults, gay and bisexual men were reported to have a higher prevalence of panic attacks, depression and psychological distress than did heterosexual men while lesbian and bisexual women were reported to have a higher prevalence of generalized anxiety disorder than did heterosexual women. So, yes some studies suggest that there are some mental disorders which are more frequently reported in the LGBT community. Of course, there are also studies which do not report such discrepancies. The more important question is: Why is the prevalence of some mental disorders higher for those in the LGBT community?
Populations that face discrimination, family rejection, social stigma and bullying are faced with many challenges that may increase the likelihood of a mental disorder. These are all risk factors for various mental disorders, and therefore increase one’s chances of being faced with experiencing a mental disorder. More details can be found in this study of LGB mental disorder, suicide and self harm.
Although the LGBT community has been found to display a higher rate of mental disorder than heterosexuals, it is dependent on the experiences of the individual. Do they have an accepting family? Do they have a support system of friends? Is their sexuality accepted in their workplace? Have they experienced discrimination based on their sexuality? Have they been bullied? The answers to these questions provide more information about one’s susceptibility to mental disorder than does one’s sexuality.