Does changing web hosting affect search engine rankings?

As long as your domain name and hosting country stay the same, switching from one reliable host to another should have no SEO impact. See question 4 in this Seomoz article link

No. And why should it? Your hosting has nothing to do with the quality or relevancy of your content. (1)

Just try to minimize downtime and avoid if possible. A seamless transition is ideal.

(1) Really slow loading webpages may cause google to lower your page rankings but you have to be in the bottom 1% or so for this to happen. So as long as your host isn't poor you will be fine.

In a similar vein as John Conde's suggestion (though picking up on a few things I've noticed) you can also expect problems when switching hosts if your new host has security issues.

If you share an Apache server with malicious users, you may find some problems including malicious content apparently hosted under your domain if your host has not disabled features like Apache's mod_userdir (it doesn't help to be recognized as a phishing or malware site)