Every television ad for the first four months of the year would probably be some variation on:
"Remember, the FBI does good work! Be sure to remember to contribute to the FBI on your next tax return!"
I also don't realistically believe this will ever happen.
Also, I'm not sure why making government departments beg for money and have to live in the free market is on the face of it a bad thing. Would it be any different than the current system, where politicians beg us for money while promising to allocate our taxes a particular way, but then usually fail to follow through completely on? Having the departments beg for money directly seems more efficient somehow.
Of course, I think what would really suffer is bold new infrastructure programs. People would be willing to allocate money to roads once a couple high profile bridges collapse, but would we ever see a space elevator?
(The rest of the time it'll still be ads for medications for diseases you never knew you had.)