The recent election has forced the prime minister to seek help from the DUP to make up the numbers, as she lost a few seats, and can't really form an effective government without a majority of seats in parliament. I've read a few articles about the DUP, and they seem like bad news for almost everyone (mostly women and LGBT people). I'd suggest reading up on them, even if you're not in the UK. Long story short, they've repeatedly blocked equal marriage, said some pretty bad things about women, they're trying hard to outlaw abortion and think that just because you've got strong religious beliefs means you shouldn't have to abide by equality laws. If you are in the UK, especially if you're in a conservative constituency, it might be a good idea to write to your MP asking what they're going to do to protect and progress LGBT+ rights, and what they will do to stop these right wing religious lunatics from dragging our country back to the 1970's.
Ruth Davidson, the Scottish Tory leader who is gay, has got reassurance from Theresa May that LGBT rights in the UK will not change because of this coalition. Plus, most people in the UK support LGBT rights, and during this turbulent period I can't imagine the Tories doing anything that may decrease their popularity even more. By all means read up on the DUP (I'm not very keen on them myself) but I wouldn't get too concerned about 10 MPs.
I'm inclined to agree with reciprocal, I think the coalition will mainly focus on brekit (one of the matters they agree on), they'd be too divided on abortion and gay marriage. Even if they forced the tories hand, they'd be completely blocked by the shadow cabinet and public opinion.
I follow politics quite closely, I don't think there will be any harm to LGBTI people outside of Northern Ireland. I doubt anything will change in Northern Ireland either - which is depressing regarding rights. This could, of course, all change dependent upon the current government's ability to maintain its power. I believe the next few months will show if they can maintain power.
The real risk is whether they'll try to punish Sinn Fein and lead to unnecessarily increased tensions in NI, not whether they'll hurt LGBT people, which the Conservatives have guaranteed not to do. The 2015 proposals they made for a targeted removal of funds from non-sitting MPs (Sinn Fein's practice) and for preventing foreign financing of parties specifically to prevent Irish and American money from going to Sinn Fein would be ill-advised moves for the Tories. Those are legitimate reasons to raise concerns about their role in propping up the government, more so than LGBT rights.
I don't think there will be any change to legal protections for the LGBT+ community on the UK mainland. If power sharing breaks down in Northern Ireland though, it could have some ramifications for the DUP, because direct rule may remove exemptions that existed under the devolved arrangement. I'm not entirely sure about this, but I can imagine it being a possibility. At this stage, I'm more concerned about what will happen in Northern Ireland as a result of any arrangement between the Conservatives and DUP. It's going to be very difficult for the UK government to act as neutral intermediary when it's being propped up by the increasingly belligerent DUP.