I used the "reissue certificate" functionality at a SSL certificate vendor (RapidSSL, FWIW) to get a new certificate - in doing so, I created and used a new private key and pass phrase.
Will the re-issuance of this certificate cause the previously-issued certificate to become invalid? If so, how long does that take?
Not automatically, for RapidSSL. For other vendors and/or certificate grades, it may be be automatic.
RapidSS don't automatically invalidate a certificate when it is re-issued, according to their Certificate Practice Statement. This would be a function of how much you pay for the certificate.
In section II.B.5 of the currently-google-able CPS:
In section III.I it says:
Elsewhere it minimally promises that revoked certificates will be added to the CRL 'at least once a week'.
Reading the Certificate Practice Statement of any purchaser of SSL certificate services is a good thing for a buyer to do.
Yes, they will revoke the old certificate.
The way that SSL revocations works is that, inside the certificate, the vendor places a URL where the client (e.g. browser) should check if the certificate is still valid (called a CRL).
So there is no hard and fast answer to this, it depends on the client. In some cases, like this article, it suggests it won't be checked at all.