Can I run fsck on a partition which is unmounted but on a live machine?

Sure, that's no problem. It's not an uncommon maintenance operation, you just need to make sure everyone knows what's going on (so they don't try to use the partition), and any services that use that mount are stopped.

Assuming you don't have odd block device permissions, nobody will be able to access and modify the block device (partition) itself -- so you can't damage the filesystem. There could, in theory, be issues with users trying to do things with the mount point while it's unmounted, but there shouldn't be any files in there, and if the mount point's permissions are what they usually are (root:root), nobody (other than root, which shouldn't be a problem) will be able to write to the mount point.

So, the only remaining possible problem is users trying (and failing) to write, and then calling the helpdesk (which you can manage with proper notifications) and automated processes and services attempting to write, which you manage by disabling those while the fsck is going on.