If I revoke an existing distribution certificate, will it mess up anything with existing apps?

I built an iOS app for an organization that has an app already on the store. After weeks of trying to get the guy who has the key to sign the app, they finally came back and said, "Just get it done!". So I am wondering how to proceed. If I go into the provisioning portal, and revoke the dist certificate, and then re-assign one, will I then be able to sign the app and upload it without problem?

That is what I was going to do, but I don't know the ramifications for the existing app. Will it mess anything up with that? And then when the organization wants to continue updates on their apps, can't they just revoke, and then reassign the certificate to them again?

This part of the process is a bit foggy to me, so a little clarification would be appreciated!!

There is no problem doing this unless you are on an enterprise account. Distribution certificates expire anyway, so eventually it will happen that you need a new one. Go ahead and delete away.

You can also find this question asked, answered, and asked again many times over on the Apple Dev forums (e.g. here's one), so google around there if you're still hesitant.

About Enterprise Developer accounts: With thanks to Mike's comment

An App store app gets resigned with an Apple certificate when it goes on the store. Revoking the cert in the provisioning portal therefore won't affect it. Enterprise apps use the original certificate, which means revoking it will cause the app to stop functioning on all devices it is installed on. If you revoke an enterprise account's certificate, all apps installed on all employee devices will stop working

Revoking a certificate has no relation to the App Store or existing apps. Once you revoke your certificate, it will be deleted from the list of certificates. Revocation has these effects:

1. You can no longer build apps in Xcode using provision profiles containing the revoked
   certificate.

2. You can no longer submit apps to the App Store that were signed with the revoked certificate or built with the affected provisioning profiles.

You can revoke it after you have paid for your next year of service. It will then prompt you for a new certificate. You submit your CSR, download the new cert, and remake your provisioning profiles.