If you take a dosis that will produce a fullblown psychedelic experience you are very unlikely to fall asleep at all for the next 5 to 6 hours.
If you, however, take a significantly lower dose than that required to produce a full on psychedelic experience, then you might be able to fall asleep on it.

Sleeping at the end of mushroom trips, when I was no longer on the peak but still very defenitely under it's effects, I have often had extremely, vivid, realistic and well remembered dreams.
Same, or actually even more so for LSD.

Taking 1 tiny mushroom some 30 minutes before bedtime might do wonders for you.