Not in my experience they don't. I use OVH and nfoservers and I've had an issue like this exactly once on both hosts.

On OVH one of my servers was hacked and running typical scripts that are run once that happens (port checking, common admin credentials, brute force attempts, etc)

They cut off all internet access to and from the server and sent me an alert stating what was happening and that I needed to VNC into the server, resolve the issue, and let them know how/why it happened and how I resolved the issue. Once that was done they just removed all the blocks on the server and we all went on our merry way.

Edit: To clarify the VNC console is on their site, not a remote connection.