I'm completely serious, do I still need to file my taxes? It appears every federal agency is being defunded and unstaffed.

Yes, you still need to file  your return.

Yes---the IRS still exists.  If you have taxable income then you need to file your tax return and stop being overly dramatic about it.

Who has to file?

http://www.irs.gov/uac/Do-I-Need-to-File-a-Tax-Return%3F

I would say yes.  There is a lot that the President can do by executive order--mainly changing how the executive branch is managed and how executive departments prioritize their activities--but the President can't single-handedly nullify a law, such as the tax code.  At least for now, income tax returns are still due April 15.