Can I prevent Piwik from making any requests to the piwik.org servers such as when checking for the latest version?
No that is not possible. Why do you want to block it?
Edit: Correction - you should be able to disable it by deactivating the CoreUpdater plugin.
Why do you want to block it?
For privacy. 
you should be able to disable it by deactivating the CoreUpdater plugin.
CoreUpdater is not listed under Plugins Management but the dashboard always tells me automatically when there is a new version.