Question: Should you be removing "bad" backlinks to your site?
This is a question I get asked a fair amount and there unfortunately isn't a definitive answer. The reason there is no solid answer is that as always not all websites are created equal.
Lets first look at where the question comes from.
The Google "Bad Link" Update / Penguin
Penguin as an update grappled with killing web spam, this included wiping out the value of "unnatural links" - which incorrectly people read as irellevant, sitewide and spam links.
"Unnatural links" are those which are created by non-organic means or which appear thousands of times, so things like paid links and links with the same anchor text (thousands of times in an effect to spam or trick google) are not natural. Yes, in many cases this can involve them being irrelevant, sitewide and spammy but not always - indeed I find the correlation to be quite low.
So should you remove these "unnatural links"?
This is the billion dollar question. Given the spate of link warnings Google Webmasters is issuing, I would say you should simply be more aware of what you are doing.
Low quality links which you've created with the same anchor text across many websites and blogs is unnatural and the links have been devalued. It's the same with article websites, press release websites and some other website types (eg classifieds). The answer to the question is that they have already been devalued, yes they maybe so again but you've already been hit by the update through them. So in this case no you shouldn't put your effort into removing the links.
The real question is what should you be doing post-penguin and post-panda?
The answer is really simple.
If you get a notice for "unnatural linking" take a look where it could be coming from. If it's scraper sites send google and bing a spam notification / take down notice. Usually within a few days they will remove the site from listings. This also shows you are aware of the issue and working on it. This is a much easier way than contacting a webmaster of a scraper site in most cases. If the unnatural links are within an article or someones blog post go back and ask for a "nofollow" or to remove the link - but don't put too much time and effort into this, it's not worth it.
Stop looking outside your box. Your website is your passion it should be number one on your list of websites to care for, so do it. Create fresh relevant content whenever possible, ensure that sites you link to are relevant and ensure that any navigation you have is easy to use. Also make your content accessible to everyone.
Stop looking outside your box. Make your website perfect before looking to the outside for help. People won't link to your site unless it feels good and full of great content - your website should feel trustworthy and updated on a regular basis.
From now on only build links inside your container, niche. Don't go off leaving blogs on SEO blogs, like SEO Andy, unless you are in the marketing industry. If you are a shop you have a clear niche to target - so whether its forums, blogs or reciprocal linking - do it in your niche.
Finally, always remember quality over quantity. This is both for your own website content and creating links. Think carefully about what you are saying and doing, it reflects on your business.
If you are still stuck on removing links, check out this article by distilled on removing bad links.