Is it important (or necessary) NOT to use an article in this sentence?
Yes, if you'd say "Egy bírót nem festek.", that would mean that you will not paint one judge, but you might paint two.
Is this an innuendo for a bribe, or just a silly sentence?
Is it correct to have "nem bírót" instead of "nem festek"?
I think nem bírót would imply you're painting something else.
Yeah, that makes sense. If you are painting "not a judge" there must be something else you are painting since you said festek.