Editing a specific custom field in JIRA can't trigger this kind of change natively. You might be able to use a workflow post function and a plugin like ScriptRunner in order to bring about a change like this, but even then it would only happen on an issue transition.
There is a related post on this over at https://community.atlassian.com/t5/Answers-Developer-Questions/RemoveIssueLink-Post-Function/qaq-p/525086
that might be helpful.
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