If a YouTube video is posted on Facebook, do the views still count on YouTube every time the video is accessed through Facebook?
YouTube counts views regardless of what site the video is embedded in.
One of the analytic reports they offer is Playback Locations which details which includes the following metric (amongst others)
Embedded players on other websites: This shows how many people viewed your video when it was embedded on another website, like a blog. Click on the link to drill down to a breakdown of views across the various sites that have embedded your video.