First off, the obvious: errors don't make a good impression on people who visit your site. They undermine the confidence visitors have in your content. Excessive errors increase bounce rate, discourage people from sharing your content, and disincentivize people to link to your site.
The BBC reported on research by Charles Duncombe, an online entrepreneur, which showed that revenue was twice as high after an error was corrected and that a single spelling error on ecommerce websites can cut online sales in half. This incredible conclusion was calculated by measuring conversion rates on ecommerce sites before and after spelling errors were fixed.
When online matchmaker OK Cupid studied first contact response rates between members, they found that "grammatical and punctuated" messages resulted in a 37% increase in positive reactions. Even those not seeking love agree; 80% of people think that good spelling, grammar, and layout are important for online copy according to Search Laboratory.
For those interested in SEO, errors also have an impact there, too.
In Google's Webmaster Central Blog, they ask the question, "What counts as a high-quality site?" and they say, "Below are some questions that one could use to assess the 'quality' of a page or an article. These are the kinds of questions we ask ourselves as we write algorithms that attempt to assess site quality."
Those questions include:
We noticed a while ago that if you look at the PageRank of a page - how reputable we think a particular page or site is - the ability to spell correlates relatively well with that. So, the reputable sites tend to spell better, and the sites that are lower PageRank, or very low PageRank, tend not to spell as well.
[W]hy would an engine show a page of content with errors higher in the rankings when other pages of error-free content exist to serve the searcher? Like it or not, we're judged by the quality of the results we show. So we are constantly watching the quality of the content we see.
One interesting case study is Zappos, which hired editors through Mechanical Turk to remove spelling and grammar errors in reviews left by Zappos users. Zappos claims to have seen "substantial" increases in revenue.