I agree that professional with the public is the way government should typically act. However, there is room for “business casual” as well. If it’s done within the context of what the organization already does…like poking fun at itself…but yet still contains some serious information, I don’t believe the public would mind that.
And to clarify, we’re not talking about changing something like the IRS homepage, or WhiteHouse.gov with a completely “joke” page. You definitely don’t want to mess with what people are trying to accomplish by coming to your site for serious work/business.