Some platonic queer relationships involve sex. Some do not. The same goes for kisses or other signs of affection. However, it is generally more than just an "extreme" best friend, even if it doesn't seem like it on the surface.
My queer-platonic relationship is not romantic (I myself am aromatic), but it is far from the best friendship. We definitely consider ourselves partners, we hug a lot and occasionally share a kiss. There is no sexual element.
It's not the only way to have a QP relationship, but it works for us, and that's the real point of an unconventional partnership.
Basically, we are in the relationship that we are because we are not comfortable with the expectations that accompany other labels. "Boyfriend" and "girlfriend" don't exactly represent him, and we feel too close to be called friends. Perhaps most importantly, as an aromantic asexual, I am not really comfortable with other types of relationships.
Actually, a platonic queer relationship can be anything you make of it, but I would call it an alternative society without limits. The main idea is that the partners feel comfortable in it.