This is entirely up to the issuing CA's process. Thawte, for example, happily revokes-and-reissues certificates for free (perhaps only for "enterprise" customers?), and the newly issued certificate has the same end-validity date as the revoked certificate but the start-validity date is set to the time of issue.
I notice herokuapp.com's CA is DigiCert, so perhaps they have the opposite policy, of giving the reissued cert the same start date as the revoked cert.
I don't think there's a standard field in an X.509 cert for issue date.
It's possible to download a CA's CRL and look for revoked certs, but all you get are serial numbers and revocation dates, not subject names.