I can't speak for prior years, but this is HUD's contigency plan for this shutdown:
portal.hud.gov/hudportal/documents/huddoc?id=HUDContingencyPlan2013.pdf
From a National Council of State Housing Agencies press release,
During the briefing, HUD officials confirmed that Section 8 Housing Assistance Payments (HAP), contract renewals, and administrative fees for October 2013 will be paid on time, due to advanced appropriations allocated for this purpose.
This clearly doesn't answer your question about future months, though. According to this article, during the 95-96 shutdown, "Local HUD officials reported that their agency had enough remaining funds to pay Section 8 rent vouchers through the end of December, but there was some question about the status of the January payments." Given that there were two shutdowns, the first partial shutdown occurring in mid-November, and the second at the end of December, it isn't clear if the remarks were made after the first - in other words, they had enough for the following month.
It seems to follow from that experience, as well as from what HUD is saying now - don't count more than one month ahead.