You may be very passionate about equal rights for LGBT people. You may even be gay yourself. Whether you are LGBT or an LGBT ally, you should not place that on your résumé, because doing so will severely limit your chances of employment. Homosexuality is still a controversial issue, despite that same-sex marriage has been legalized in all 50 states of the United States; and openly LGBT people are struggling to achieve access to jobs, adoptions, access to business services, and others.
I am not LGBT, and I am not involved with LGBT activism. It is not that I do not approve of the rights of LGBT people. It is just that I would risk my own life if I support a politically controversial cause that does not even apply to me in any way, shape, or form, and receive no benefit from doing it. There was one time on the college campus when one guy approached me on the street, asking people to sign up for LGBT rights. I politely turned it down by saying I didn't have any money. That was true; I was not employed. Why would I support a cause for people who might be my competitor?
Plus, I have always been apathetic to politics in general. My paranoia with expressing strong political sentiments extends to non-activism cases too. I remember during my undergrad years, I saw a handful of itinerant college preachers who caused angry uproar among the young adult crowd. I supported neither side; all I did was silently watched for a moment and walked away. Naturally, I get suspicious and skeptical with overly friendly religious groups. I may be more open to well-established religious groups, like the Catholics, Orthodox, Protestant groups that have been around since the 1700s, Jews, and Muslims, but non-denominational evangelical Protestant groups, obscure religious sects, controversial Christian sects (Jehovah's Witnesses and Latter-Day Saints), and atheist groups strike me as "too political and politicized".
Therefore, last summer, I attended Catholic Masses and attempted to be active, so people would notice me, and I might network with them. It was a good experience, as I gained a new personal reference. However, the more I got involved with Catholicism, the more I realized it was not for me. Catholicism may seem like an ancient, mainstream religion, but listening to the Catholic radio station just gives me a headache. That radio station is peppered with politics, religious hypocrisy, and sensationalism.
As a rule of thumb, I would avoid inserting political activism, especially LGBT activism, on your professional résumé. Some political activisms may be more acceptable than others, such as feminist activism (very mainstream) or environmental activism (a bipartisan issue), but I believe it is best to avoid placing them on the résumé, because you'd never know whether the hiring manager has a problem with that.
LGBT activism, on the other hand, is associated with homosexuality or rejection of gender norms, and so it will be taken more gravely. Additionally, you may consider making different résumés for different job positions. If you want to work for an organization that assists LGBT people, then LGBT activism in your personal history would be a strength, as that may indicate passion and dedication to the cause. That may be the only place where listing LGBT activism is acceptable.