I've no idea what he said on this subject, haven't seen any video about it, and this topic didn't come up in the few talks of his that I went to.
But - whatever he says - the guiding principle is to act out of wisdom and compassion. It's not really a doctrine based religion; the idea is to think things through for yourself.
You don't have authorities like the Pope. The Buddha in his teachings particularly said not to take anyone on as any kind of a head of Buddhism after he died.
The Dalai Lama only speaks for himself and advises his own immediate students. He doesn't speak for all Buddhists. He doesn't even speak for all Gelugpas. He can't, because Buddhist teachings don't work like that.
You are expected to not accept anything as true on anyone's say so, except in an interim way as in "I don't know, but this seems worth investigating". Anything anyone says are just hints to help you to find the truth for yourself.
There are the precepts, moral precepts, you can choose to take - but those also date back to the Buddha's time, and can't be added to by present day Buddhists, and the Dalai Lama or anyone else can't change them, can only give their interpretation to help you to find out how best to keep to them.
There is no such thing as a Buddhist "decree" such as the Pope can issue.
As for this question - whatever he said - the motivating thing behind it all should be compassion. If you have a compassionate heart and wisdom, you'll find the right path on issues of the day.
The sutras are the nearest there is to a doctrine in Buddhism - but not in the rigid way as in this is what you have to do. I don't know if there is anything in them on this topic. Again that's a bit different from Christianity. Many Christians will have read the bible or new testament, or at least the gospels in their entirety. But the Buddhist sutras are so vast, only scholars have read them all. It's an entire small library of books and it would take ages just to have a first read through of them all, while indeed Buddhist scholars can spend years studying just one of the sutras and its later commentaries.
Some people actually do not only read, but memorize the entire Buddhist Tripitaka - it's a 16,000 pages in the Pali canon - and the examination to test if you've memorized it correctly - and with understanding also - is the longest examination in the world, and lasts 33 days at ten hours of examination time a day. Not surprisingly the number of people who have done this is rare. MingunSayadaw
Also interpetation of the precept on sexual misconduct depends on your culture e.g. some Buddhist cultures practise polyandry or polygamy etc, the guiding principle is to avoid engaging in sexual activity that is harmful to others (or to yourself).
So what would count as infidelity in one culture according to the precept - because your partner would expect faithfulness to them and them alone for life - wouldn't be infidelity in another culture or in the same culture but a different sub culture and expectations.
If you lead compassionate, loving and fulfilling lives, and bring happiness to others and don't cause any harm to others, bearing in mind what their expectations are in the society you are in - that's the main thing.
That's more important than the precepts actually, especially in the mahayana traditions such as Tibetan Buddhism. Or rather, that's what the precepts are there to help you to do. But the compassion comes first, not the precepts.
There are plenty of stories in the teachings I went to about buddhist practitioners in the past who broke their precepts out of compasssion.
So interpretation of the precepts depends on society, in the details of how they work, and compassion and wisdom ultimately trump the precepts.
I don't know what the Dalai Lama said or didn't say - but you do get lots of these questions here on Quora saying that "the Dalai Lama said such and such" and usually without any source and especially not with a video or a transcript or book reference so you can see the wider context of what he was saying.
Whatever he said, then the guiding principle is compassion and wisdom - and that over-rules anything your teacher says also - finally - there is no way that your teacher - even if you go as far as to take on a "guru" (very few Buddhists do) - still - he can't tell you to do anything that goes against your own inner compassion and wisdom - if he does you rightly ignore it, though of course there's plenty of scope for false compassion and wisdom where you think you understand and don't.
And anyway - it's not a prescriptive religion either - so - the people who need to know the answer to this are Buddhists who are gay themselves in order to decide how to behave - and they have to find their own answers to what they consider to be compassionate and loving and wise in their situation - what counts as appropriate behaviour and what counts as sexual misconduct for them - as someone who isn't gay myself - all I can say is - that many gay people surely live happy and loving and fulfilling lives, and are thoughtful and wise and considerate in their actions, as best I can tell. While others doubtless do engage in sexual misconduct, as do many straight people.
Oh - and if you live a life aimed at maximizing sexual gratification (e.g. like people who have lots of partners and as soon as they get slightly bored of one move on to another with sexual satisfaction as their only reasons for their decisions) - then even if it's done without any cheating and with consent etc - that's going against some others of the Buddha's teachings, this time, the teachings on refuge. So if a Buddhist finds himself or herself in a situation like that would probably be an idea to examine your life choices and think a bit about where you are headed.
Because - ultimately - that's a refuge that's going to let you down - at some point you will find you can no longer solve your problems in that way - and it won't ultimately free you from suffering, or help you to find true happiness. Which I think is reasonably clear to most people, that it's not like a final total solution to all your problems in your life to find a congenial sexual partner or partners (or indeed friends either)- so again that's the same whether you are straight or gay.