Adding the nofollow attribute to links prevents PageRank from being passed. This is something that Google wants webmasters to do for any links that have been purchased. To do otherwise is strictly against Google's quality guidelines. Violating these guidelines can either get you hit with a manual penalty in Google, or get you snagged by Google's Penguin updated, which will continue to see regular data refreshes.
Some webmasters have wondered if having a large amount of links with the nofollow attribute pointing to a page could hurt that page in search. Barry Schwartz at Search Engine Roundtable points to an interesting Google webmaster forum discussion, in which Google Webmaster Trends analyst John Mueller sets the record straight.
In that thread, user rickweiss writes, "Bloggers have apparently taken the issue of never having a dofollow on any link that is tied to something you are compensated for so seriously that they are putting nofollow on all links in their posts. In other words, the legitimate link to the products page is getting a nofollow."
Later, a user going by the name Bens Mom, asks, "I am incorrect in the belief that having too many rel=nofollow links can actually hurt a site? Because that is the impression I'm under."
I'd like to back up what others said -- having links (even a large number of them) with rel=nofollow pointing to your site does not negatively affect your site. We take these links out of our PageRank calculations, and out of our algorithms when they use links.
If you've been doing this for a longer time, then it might even make sense to work to clean up those older links, and to have them either removed or the rel=nofollow attached (given that those kinds of paid posts would be against our Webmaster Guidelines).
This isn't much of a surprise, considering that nofollow is designed to do what its name would imply: keep the search engine algorithms from following these links. That would indicate that these links carry absolutely no weight one way or another.
Google's constantly changing algorithm has a lot of people paranoid about their linking strategies, and it seems that some are so worried about Google's actions that they're taking unneeded actions of their own, and ironically, possibly hurting SEO in the process.
Image: John Mueller, from Google+