I have found a site that is stealing all of my post from my blog. I'm working on fixing it, but in the meantime am wondering if it will effect my links or get me banned form any affiliate program.
It could get you a duplicate content penalty for SEO. Wouldn't matter with affiliate marketing unless they are adding their affiliate links. Then they would get the commission and not you.
The site that is copying is supposed to get duplicate content penalty, not you. Most likely google will be able to identify the original from the date at which article appear, the trust and importance of both sites etc. if your site is overall better ranked and having a better PR, you will not have to worry about it.