The biggest problem I can see with this is inefficient resource allocation. Others have mentioned ways of giving money to yourself, but we could probably minimize that with conflict-of-interest controls or by scoping budgetary buckets correctly. But there's no reason, even in principle, to think that the public's willingness to donate to a government office corresponds usefully to its actual needs.
As a toy example, let's say the public really likes puppies and decides to put, say, 1% of GDP into puppy shelters and puppy-related veterinary programs. Diminishing returns kick in at 0.1% of GDP; puppies are still being saved, but at that point marginal dollars would be doing more good directed at kitten shelters (which were too busy herding cats to spend time on outreach in the run-up to tax season). The last puppy is saved at 0.5% of GDP, and the remaining 0.5% -- after a modest indirect subsidy to casinos and makers of exotic sports cars -- goes into the newly minted Bureau for Puppy Salvation's public education fund.
Next tax cycle, that investment pays off and puppies get 2% of GDP.
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