This is an official answer from someone who works in the ticket booths at DLR.
"When you renew online, your old pass becomes void, you can only have one active pass at any time. you have to wait for your new pass to arrive or go ask for a courtesy pass at the main gate ticket booth" (aparently this is a common problem that they fix all the time - you might as well renew at the booth)
Thanks for the clarification. Technically, you only get the renewal discount and gift coupons if you do it online, but I have gotten both the discount and the special coupons by renewing at the park.
If you get discounts online then by all means, do it online. I'm sure the passes show up pretty quickly, I know they are really concious of people waiting in the lines at the main gate, I'm sure they want the system to help eliminate the waiting times overall at the park.
I have renewed online, and carry a printout of my payment with my AP until the new one arrives. Usually the Cm will override the system.
But that was priopr to the Monthly payment plan. You might now have to go to Guest Services if the AP is not accepted at the turnstiles. But I would try the turnstile first, if it is a week or so before the expiration date, usually the CM will ask if you have renewed online, as it happens on a regular basis.