I haven't thought about this super much, but I think it would be bad for people to think that you were "officially" affiliated with LessWrong.
It seems totally fine to mention LessWrong in the video description, or in your videos, or comment over footage of using the LW website, but using the LW logo or name like a news-report might use their channel logo would probably be confusing for people.
I also don't really have the final say in this. The LessWrong brand is still owned by MIRI, though I expect that they would roughly give the same response as I would.
Feel free to comment with a more precise description of what you were hoping to do with the LessWrong branding, and I can give you a more definite yes-or-no depending on the precise use-case.
+1 on not wanting people to be confused about the relationship between the youtube channel and the site, and a channel called 'LessWrong' that wasn't by someone who was at least a major writer on the site would lead to confusion.
I am generally pro marketing not by name but by example. Living the principles rather than cheering them. Lots of videos asking people why they believe what they believe, spending the time doing scholarship, asking the questions most likely to change your mind, and all the rest, as opposed to overly talking about how great rationality is or using lots of buzzwords.
Anyhow, thanks for asking - and best of skill with the videos! (Drop a link here once you've put some up.)
The most indiscrete example I have in mind would be to include it in the channel name; a clunky example would be 'Less Wrong Street Epistemology'. I had no intention of using the official logo, merely make direct references.
From my perspective the term LessWrong implies community and a nonprofit way of doing things.
Given the nature of LessWrong I don't think that kind of branding you propose would be helpful to LessWrong as controversial things you do on your channel get attributed to LessWrong if you have LessWrong in your channel name.
There's a cost to doing controversial things. Our local LW meetup for example lost a room because the LW website allegedly talks about "race and eugenics".
On the other hand, there's little potential for harm if you just link to LessWrong.
I will be starting a youtube channel where I record my Street Epistemology encounters. I want to heavily plug our community, for the benefit of growing it, and also because it's just so darn appropriate (I am literally asking people what they think they know, and how they think they know it).
I will not be claiming ownership of the branding (I'll be diverting people to the website, of course), but I would probably go so far as to use it in the name of the channel (for lack of a better pithy name, 'Less Wrong' is excellent). Are there any legal problems with this?