I have a site that auto-posts to twitter (and facebook) different content from my site (always unique) about 5 times per day. Will including a backlink to my site in every post negatively (or positively) impact the SEO of my site?
It shouldn't hurt SEO to tweet links from your site. This site has a twitter bot that does so. The NY times tweets all its stories.
Google usually does monitor the twitter stream. It uses that data to discover content. Tweeting a link from your site to a new article has been a good way to alert Googlebot to the link before scrapers can steal it. I haven't seen evidence that Google uses twitter links to pass PageRank. Twitter links probably don't help rankings in any way.