If we look at the canonical quote listed in the answer to "Does Jack Harkness "need" to eat?", courtesy of Rand al'Thor:
DOCTOR [behind door]: When did you first realise?
JACK: Earth, 1892. Got in a fight in Ellis Island. A man shot me through the heart. Then I woke up. Thought it was kind of strange. But then it never stopped. Fell off a cliff, trampled by horses, World War One, World War Two, poison, starvation, a stray javelin. In the end, I got the message. I'm the man who can never die. And all that time you knew.
We see that he can die of a sickness that any mortal man can die from, including starvation. Since sleep deprivation is actually lethal (what was the record, 6 days?), we can assume, although it is not documented, that Captain Jack can die of lack of sleep, and thus regularly needs some.
But again, I have no canonical evidence.