It depends on the site. Have I been there long or am I new? If I'm new and I feel that the warning was given out of bias on the moderator's ideals then I move on because that site is not worth my time or effort. That happened not too long ago.
If I've been on the site for a long time, then I determine if the warning was warranted. I might stick around if they can prove that I was in the wrong and I enjoy posting on the site.
Generally I do not receive staff actions because I try my best to follow the rules.
As long as the Moderator can justify how I broke the rule then it's fair enough, I would carry on posting on the forum if I like it there. If not, just leave the forum or if you really have nothing else to do, take it to social media to let everyone know how bad that community is. Nothing hurts a forum than having a bad reputation, I guess.