The user lordazoroth originally posted this method here in the T-Mobile SGSIII forum. However, I have confirmed that it works with the Note 3 (at least with the Verizon variant) with one small difference. Feel free to give us both a "THANKS" if this helped you out.
You can disable the OTA Update notification by following the steps below. Just be aware that there is a small change in STEP #3.
Step 1. Delete the update located in /cache/fota/ (may be /data/fota) and then click the update notification prompt to install it, it will instantly fail and the notification will disappear. (My phone rebooted and tried to install the update - this is the point where my Note 3 gave the update failed notification. I just rebooted the phone and went to STEP #2.)
Step 2. Disable otacerts.zip located in /etc/security by renaming it to otacerts.zip.bak
Step 3. Disable FotaClient.apk and FWUpgrade.apk by renaming with .bak extensions located in /system/app/
Step 4. Reboot
(NOTE: The difference in STEP #3 for the Galaxy Note 3 is that the [FotaClient.apk] is named [LocalFotaClient.apk] instead of just [FotaClient.apk]. All of the other steps are the same.
Continue running JB 4.3 and enjoy your Note 3 without that nagging update notification!
confirmed for samsung galaxy note pro 12.2 on November 22nd, 2014.