Can I run HTML files directly from GitHub, instead of just viewing their source?

There is a new tool called GitHub HTML Preview, which does exactly what you want. Just prepend http://htmlpreview.github.io/? to the URL of any HTML file, e.g. http://htmlpreview.github.io/?https://github.com/twbs/bootstrap/blob/gh-pages/2.3.2/index.html

Note: This tool is actually a github.io page and is not affiliated with github as a company.

You might want to use raw.githack.com. It supports GitHub, Bitbucket, Gitlab and GitHub gists.

GitHub

Before:

https://raw.githubusercontent.com/[user]/[repository]/[branch]/[filename.ext]

In your case .html extension

After:

Development (throttled)

https://raw.githack.com/[user]/[repository]/[branch]/[filename.ext]

Production (CDN)

https://rawcdn.githack.com/[user]/[repository]/[branch]/[filename.ext]

In your case .html extension

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raw.githack.com also supports other services:

Bitbucket

Before:

https://bitbucket.org/[user]/[repository]/raw/[branch]/[filename.ext]

After:

Development (throttled)

https://bb.githack.com/[user]/[repository]/raw/[branch]/[filename.ext]

Production (CDN)

https://bbcdn.githack.com/[user]/[repository]/raw/[branch]/[filename.ext]

GitLab

Before:

https://gitlab.com/[user]/[repository]/raw/[branch]/[filename.ext]

After:

Development (throttled)

https://gl.githack.com/[user]/[repository]/raw/[branch]/[filename.ext]

Production (CDN)

https://glcdn.githack.com/[user]/[repository]/raw/[branch]/[filename.ext]

GitHub gists

Before:

https://gist.githubusercontent.com/[user]/[gist]/raw/[revision]/[filename.ext]

After:

Development (throttled)

https://gist.githack.com/[user]/[gist]/raw/[revision]/[filename.ext]

Production (CDN)

https://gistcdn.githack.com/[user]/[gist]/raw/[revision]/[filename.ext]

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Update: rawgit was discontinued