According to the excellent UKCampsite.co.uk, there is only one campsite which allows fires inside the New Forest park boundary. It's this one:
http://www.hurstviewleisure.co.uk/open-camp.ht m
There are 29 reviews of the site here:
http://www.ukcampsite.co.uk/sites/reviews.asp? revid=3950
If you're going to be looking for sites which allow campfires, elsewhere in the UK (or even on the continent), you might like to bookmark this page:
http://www.ukcampsite.co.uk/articles/view.asp? id=4
Chris