if you will edit created test plan with assigned test suites, and change assigned person - the changes will not affect to the test plan.
Thanks for your posting. This is actually not a bug but the intended behavior when you edit a test plan (mainly because updating the assignments would override possible manual assignments you may have configured before). The Assign To field is currently only used for new test runs for the test plan entry (when you add configurations, for example). We have plans to improve the test plan form (and the assignments) in one of the next TestRail updates but for now I would suggest using the Mass Assign feature to change the assignments for existing test runs.
This is also explained in more depth in the following thread: