Let's say you hosted a government website with over 1 million visitors monthly. You'd probably want to monetize it. What about considering Google AdSense? It's a possibility; the contextual ads are likely to be well-targeted. Thousands of websites employ Google AdSense to make some money off their content, so it's not unheard of to try this in the government sector. The question, though, is what happens if some sketchy ads from undesirable businesses or bogus sites go through on the website? It may look like an endorsement that the government doesn't want to be affiliated with.
Very few people in the thread suggest that it will work. "Be inconspicuous about it," one forum member recommends. Most say that it's offensive to add AdSense on sites that are already being paid for by our tax dollars.
Others don't want government sites to run AdSense at all.
If governments start running Adsense on their websites, they obstruct fair competition and the free market.
Similar sentiment is echoed throughout the discussion:
The government should be independent and therefore cannot put any ads on their site promoting either 'store A' or 'store B' nor advertise for blue or green widgets.
Do some people want it? Only if it reduces our taxes and funds education. That's very likely not going to happen, so they'll probably never say yes.
There's always the Ad Review Center which integrates a competitive ad filter for ads that you don't want to endorse. There's likely to be a lot of overhead involved for a government site, though.
Forum discussion continues at WebmasterWorld.