Get CI to build docs automatically on pull requests

This issue has been migrated from [#10997](https://github.com/matrix-org/synapse/issues/10997).

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For pull requests that change `/docs` it would be nice to have an action which

- builds the documentation book
- uploads it as an archive
- does not push to GitHub Pages

There is an action [here](https://github.com/dorny/paths-filter) which might detect a change to `/docs`. But a neater way might be to use [`on.pull_request.paths`](https://docs.github.com/en/actions/learn-github-actions/workflow-syntax-for-github-actions#onpushpull_requestpaths).