A while since I worked with tar, but I don’t think even current linux versions allow you to ignore this error.
Really you have two choices, either limit backup to exclude files which could change during course of backup, or shutdown services during the backup (easiest is probably to drop system to single user mode), and then restart after backup has completed.
If you are only having problem with one specific file, or directory should not be too difficult to exclude them.
If problem is always caused by same service, you could try just stopping that service alone (e.g. mail) while backup is run.