In general, there’s no harm, and considerable benefit, in letting backups run. Your backup software should act as though your duplicate is your regular startup disk and keep copying files to its normal destination as though you had restarted normally. That’s probably what you want, because if you create or modify a file while running from the duplicate, it can then be backed up.
Perhaps it has since been fixed, but I’ve seen cases where this didn’t work right. Time Machine saw the duplicate as a new drive. Thus, rather than creating an incremental backup, it wanted to make a fresh copy of all the files. The only way to do this was to free up space by deleting the backups of the original drive. And the same problem happened again when switching from duplicate back to the regular startup disk. If you boot your Mac from a clone, I think it’s advisable to give it a different drive to use for Time Machine.