This advice to not fall asleep is pretty prevalent (at least on TV). Is there a basis for it?

Not sure about strokes, but it used to be believed that a person with a concussion must be kept awake, but it turns out not to make a difference. It's still a fairly common trope in movies and on tv, though.

One source on that: http://www.uamshealth.com/?id=10724

It's useful to detect problems not prevent them. One of the symptoms of interracial pressure, a significant risk with head injury, is drowsiness. But, other symptoms are also fairly easy to detect in a responsive patent. However, in cases of significant head injury this can be directly monitored allowing the patent to sleep.

http://www.nlm.nih.gov/medlineplus/ency/article/000793.htm

PS: From you link "If the person who is injured is awake and holding a conversation, you can let him or her fall asleep as long as they are not developing any other symptoms such as dilated pupils or issues with walking. "

The most important thing about staying awake is that a more complete neurological exam can be performed. There's always ways to wake someone up, but the unfortunate part is that any treatment may alter the results of neuro tests

Would love to know this too. I have always thought it might have to do with being able to communicate with the medical staff to get a better assessment.