The health and human services budget has expired and a new one has not been passed yet. So how is Obamacare still functioning, as it falls under that part of the budget? Is it being kept alive by some sort of accounting tricks or the president exercising authority he doesn't actually have or some other method?
Conducting essential activities to the extent that they protect life and property.
A argument could be made that the exchanges and other aspects of Obamacare protect life, because they pay for services that protect life. If that wasn't the case, then those goverment agencies are prohibited from spending money that Congress hasn't appropriated under the Antideficiency Act (possible Obamacare exemption bolded below)
The Antideficiency Act prohibits federal employees from
making or authorizing an expenditure from, or creating or authorizing an obligation under, any appropriation or fund in excess of the amount available in the appropriation or fund unless authorized by law. 31 U.S.C. § 1341(a)(1)(A).
involving the government in any obligation to pay money before funds have been appropriated for that purpose, unless otherwise allowed by law. 31 U.S.C. § 1341(a)(1)(B).
accepting voluntary services for the United States, or employing personal services not authorized by law, except in cases of emergency involving the safety of human life or the protection of property. 31 U.S.C. § 1342.
making obligations or expenditures in excess of an apportionment or reapportionment, or in excess of the amount permitted by agency regulations. 31 U.S.C. § 1517(a).
Will a shutdown kill Obamacare?
No. Most of the money for Obamacare comes from new taxes and fees, as well as from cost cuts to other programs like Medicare and other types of funding that will continue despite the government shutdown.
Government spending rules are very complex and not all money is the same. For a glipse of this, take a look at OPM's furlough guidance during the shutdown. There are exempted employees, excepted employees, and "regular" employees (not sure what special title they are given). OPM's guidance will help you understand the complexities of federal law that are probably being used to allow Obamacare to continue forward:
In the context of shutdown furloughs, the term “excepted” is used to refer to employees who are funded through annual appropriations who are nonetheless excepted from the furlough because they are performing work that, by law, may continue to be performed during a lapse in appropriations.
Employees are “exempt” from furlough if they are not affected by a lapse in appropriations. This includes employees who are not funded by annually appropriated funds.
Employees who are funded through annual appropriations but are not designated as excepted are barred from working during a shutdown, except to perform minimal activities as necessary to execute an orderly suspension of agency operations related to non-excepted activities. These employees will be furloughed.
I imagine the way Obamacare funding was setup it would not be affected (probably by design) by lapses in appropriations.