Yes suffering from anxiety and being nervous can cause a persons muscles including the throat muscles to tighten up.
So if you're suffering from anxiety and your throat tightens up when you become nervous then it's usually nothing to worry about and once the anxiety ceases the throat should no longer be tight.
However it's always a good idea to seek a doctors medical attention to determine if the throat tightening is only caused by stress an anxiety because some other medical conditions can cause your throat to tighten up and close but if it only happens when you're nervous or stressed then it's likely just due to the anxiety but it never hurts to get a definite answer.