I have a free WordPress website with 1000-2000 visitors/month. I was trying to place Amazon Affiliate ad links last night and failed/flailed mightily. I would paste the links into my posts in the HTML mode, switch to Visual mode, and could see the image. However, every time I tried to save or update the post, the image reverted to the gangly hyperlink. As some suggested, I tried cleaning the links of any formatting from the copy and paste with text edit. No dice.
You should also make sure you aren't competing with your own affiliates for eyeballs. Any marketing channels you're using, such as search engines, content sites or e-mail lists, should be off limits to your affiliates. Put marketing restrictions into your affiliate agreement and notify partners immediately. It's your program--you set the rules. Or, if you prefer, you can let your affiliates run the majority of your internet marketing.