For example, could I somehow actually see the test results that would be produced by the jQuery test suite, without downloading or cloning the repo to my local drive and running them there?
I know this would basically put GitHub in the static content hosting business, but then again, they just have to change their mime-type from text/plain to text/html.
There is a new tool called GitHub HTML Preview, which does exactly what you want. Just prepend http://htmlpreview.github.com/? to the URL of any HTML file, e.g. http://htmlpreview.github.com/?https://github.com/twitter/bootstrap/blob/master/docs/index.html
While you cant do it directly in the repo, check out http://pages.github.com/.
You can either branch gh-pages to run your code or try this extension (Chrome, Firefox): https://github.com/ryt/githtml
If what you need are tests, you could embed your JS files into: http://jsperf.com/
I had the same problem for my project Ratio.js and heres what I did.
Problem: Github.com prevents files from rendering/executing when the source is viewed by setting the content type/MIME to plain text.
Use jsfiddle.net or jsbin.com to create a webpage online then save it. Navigate to your file in Github.com and click the raw button to get the direct link to the file. From there, import the file using the appropriate tag and attribute.
Note: Note for jsfiddle.net you can get direct access to the result page by adding show to the end of the url. Like so: http://jsfiddle.net/jKu4q/2/show
Or....you could create a project page and render your HTML files from there. You can create a project page at http://pages.github.com/.
Once created you can then access the link through http://*accountName*.github.com/*projectName*/ Example: http://larrybattle.github.com/Ratio.js/
To piggyback on @niutechs answer, you can make a very simple bookmark snippet. Using Chrome, though it works similarly with other browsers
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