Charles Peters in the Washington Monthly used to have fun with bureaucratic techniques to avoid budget cuts. I think he called one tactic: "closing the Washington Monument". In other words, try to cut an agency's budget and the wise bureaucrat will make sure the cut shows up in the most painful way possible. That's sort of what will happen if history repeats itself and the government is shutdown. If I remember the Gingrich days, there very quickly was some triage. Congress decided that things like defense had to continue (and we weren't involved in one war in Afghanistan then and one (something) in Iraq). And Social Security checks had to go out. And other critical activities had to happen.
One little-realized fact is that some agencies, or some activities within some agencies, are not financed by the budget and appropriations, but by user fees. So presumably passports will continue to be issued and meat inspections will continue.
But I'm probably confusing two issues: a possible Republican refusal to increase the debt ceiling; and a refusal to fund the government by passing a continuing resolution or appropriations bills. If I recall, Secretary Rubin used some financial manipulation to get around the first for a while, like raiding various trust funds by putting Treasury IOU's in place of the fund assets. The Republicans howled, but the tactic worked. There's a limit to how long that can go on. The second issue causes parts of the government to shut down. Because the Democrats didn't pass any 2011 appropriations bills, if and when the current continuing resolution expires the whole government would shut down. That's when you'll see some bipartisan agreement on funding things like DOD, VA, IRS, SSA with appropriations. What happens next we don't know.