No Registrar should require you to pay for a domain name renewal in order to initiate a domain name transfer away. Your new Registrar of choice can initiate a domain name transfer request on an expired domain name once they receive the required authorization from you. Note that you should have acquired all of this required domain name authorization information, prior to the domain name expiration. Once a domain name expires, you will be unable to make any modifications to the domain name which includes: obtaining EPP code, modifying Domain Privacy, unlocking the domain, etc.
Stil, not all expired domains can be transferred. Domains that have entered the "Redemption Period" will need to be restored/ renewed before they can be transferred. For some domain name extensions, primarily country code (ccTLD) extensions, there is no "Grace Period". They go to "Redemption Period" immediately right after expiration. Thus, most expired ccTLDs cannot be transferred at all.
It is sometimes possible to transfer an expired domain from one Registrar to a new one. However, you should always check with the current Registrar to verify that you can or cannot transfer the domain.
