Apparently they can. I'm not entirely sure how the setup works, but you can be banned for using the trainer in single player mode.
Yes it does. If you read the article that they posted about this ([url]www.cheathappens.com/article_blizzardbans.asp'>Link) you can see actual emails that people have gotten from Blizzard about this.
[Edited by Bes, 9/8/2011 11:15:17 PM]