CAP, a liberal US think tank, highlights some issues that same-sex couples face in health insurance.
One major issue: LGBT couples are usually not able to access insurance on the same terms as heterosexual couples. While insurance benefits offered to opposite-sex spouses and children are not taxed in the US, benefits to same-sex partners are, to the tune of about $1,069 in additional taxes per year per partner. Additionally, not all employers offer domestic partner benefits. It is an option for same-sex couples where both partners are employed at large firms to get two different policies. However, the small and individual markets are broken.
Another major issue that would likely be solved without any additional action is that pre-existing conditions are denied. LGBT folks have a higher prevalence of HIV and certain cancers.
The last major issue is that transgendered people have significant problems with insurance. Many companies have attempted to deny certain services, or coverage in general, to transgendered applicants. In some cases, insurers may have denied coverage for completely unrelated conditions (e.g. a broken arm) by claiming that these were related to gender transition. Additionally, gender transition services are generally uncovered. However, due to the small number of transgendered people in general, coverage is usually cost-neutral. Several large companies cover these services successfully.