'Fear not' is a very old-fashioned expression which means 'Don't be afraid'. It isn't normally used nowadays. 'Don't fear' doesn't really mean anything. You could say 'Don't fear ... something', but again, it is very old-fashioned and not used nowadays. 'No fear!' has a completely different use. It's a colloquial response, used mainly in informal US Eng conversation, to mean 'Of course not!'

the three phrases have same meaning , don't be afraid , but the last one is rarely used , you can say it's old-fashioned , the middle one , here Fear is a noun ( e.g you have no fear ( or fears) to speak publicly) . Finally , the first one is a verb ( e.g , you don't fear ghost ) NOTE : you can say no fear when u answer a suggestion or a question with No or ( certainly not) , you can answer with "No fear" instead , but this is usually informal