Using google adsense, am I allowed to put a header on my forum that states the forum is supported by adsense?
No I don't think you can do this. Also, if you did your members may be inclined to click on the ads to 'support' you & the site. You can label them 'advertisements' or something like that but I'm 99.9% sure you can't say your forum is supported by Adsense.
Just don't say it right above the ads, otherwise it's fine. You can also mention it in the newsletter that the ads pay for the hosting. Members won't mind if they like your forum, just like how I don't mind clicking on ads on DP. (I hate Shawn though, he's too rich for my blood).
You can say ads pay for your website cost, but I don't think you're allowed to say "help us keep running by clicking an ad!" sort of thing.
if you openly announce that the ads are paying for your site, it is possible that you are hinting for people to click on the ads it's best to say nothing at all
instead of saying 'This forum is supported by adsense', why dont you use Adsense ref button, that says - 'This site is monetized by adsense' ?? 100% legit and proffitable too..
And put that referral button close to the ads, the image would draw attention to the ads. Thus serving a secondary purpose. As for the statement of "forum supported by AdSense" - as always, if you are not too sure, ask Google.