Will installing Mountain Lion require a complete refresh of my Time Machine backup?

There is absolutely no requirement to start from scratch, the fact that a lot of OS files will have changed just means that the next run will be large, but the historical data you have already is good to stay around.

In fact, starting afresh is a really bad idea, because you will (assuming you use the same disk) lose all your historical files and the ability to revert to previous versions, or retrieve deleted files etc.

As things stand, keeping using the same store will allow you to even revert back to Lion if you so desired, the worst that can happen is the new OS install changes being large enough to fill your backup disk and force the removal of some of your older backups, but even so it's worth keeping with the same backup.

Mountain Lion will change about 5G of data, not including the space to backup the installer if you save that onto Time Machine (and who wants to re-download that again, even on a screaming fast network).

Other than that first backup where it moves the new files, there have been no reports of widespread issues with Time Machine behaving poorly with this upgrade.

That is not to say some people have other pre-existing corruption or unlucky failure at the same time as the upgrade and have had problems, but it's not the rule.