The checkout button does not work when security settings in IE and other browsers are set to "High." We deal with financial institutions and most of their internal networks set this to "high" due to their internal security requirements.
We're using version 2.80, but most of the behavior persists in version 3.0 as well (e.g. register and "add to cart" buttons also don't work)
This can also be duplicated on the demo site. You can click checkout there, so it gets to the billing/shipping/payment screen, but then you can't fill any of those out as everything's disabled.