My domain name has just expired. Can my registrar require me to pay for a renewal before I can transfer to a new registrar
No. Your new registrar of choice can initiate a transfer request on an expired domain name once they receive the required authorization from you. Expiration or nonrenewal of a domain name is not a valid reason for denial of a transfer request.
Note that if the registrar has already begun the deletion process on the domain name and its status shows it to be within a 30 day Redemption Grace Period, the name must be restored by your current registrar before it can be transferred.