It's a tough subject, Josie. I'm aware of mosques in the UK, Canada and the US that have taken welcoming positions, and if I'm not mistaken, at least one mosque in Canada has flown the rainbow. But Islam in general has not made the sort of progress that Christianity has with respect to theologically affirming positions. This mosque in Berlin is known as a liberal mosque, but based on their statement I'd be surprised if they went much further than a welcoming position. But I don't know, so don't quote me.
I lived in the Detroit area for a long time, and I knew a lot of Muslims because Detroit has been a center of immigration from mid-East countries like Lebanon and Kuwait, and from predominantly Muslim Bangladesh. I had many Muslim friends, particularly women and even very religiously observant women, who were perfectly accepting. Dearborn High School, which is predominantly Muslim, has an active GSA club and a very LGBTQ supportive faculty.
However, this sort of acceptance usually comes in spite of religious teachings and not because of them. Muslims in the US and other western nations can be acutely sensitive to the need to protect minorities, often because they recognize their own vulnerability as members of a sometimes despised minority. So they can be very likely to understand and value pluralism.
In my experience, and again please don't quote me because my personal experience is limited, imams are much less likely to be tolerant of LGBTQ people than everyday Muslims are.
Clearly, exceptions exist or we'd never hear about welcoming mosques, and that flag would not have flown over that mosque in Berlin.
Islam is not monolithic, and there (generally) are no formal hierarchies like in the Roman Catholic Church and big mainline Protestant denominations in Christianity. With the exception of some very influential leaders who have clout but no formal authority, there is nobody like a Pope or the equivalents of bishops. Imams are chosen by people who want to follow them. Nonetheless, Islam seems to exist in a paradigm where most imams take positions quite parallel to the most conservative of Christian leaders, defining the religion by traditional sexuality teachings: i.e., you cannot be queer and a Christian, or you cannot be queer and a Muslim.
Unlike in institutional Christianity, no large Muslim denominations teach otherwise, but then there are no Muslim denominations at all, because Islam doesn't organize itself like that.
In North America and Europe, it's possible to find welcoming mosques, and I suspect that trend will continue and grow. But in several ways of measuring, Islam is a good deal behind Christianity still.
