I came across the following article on wikiHow http://www.wikihow.com/Remove-the-Popup-Ads-in-Avira-Antivir, which gives details on possible soluitions on remove the popup adverts in Antivir, I belive this will break the terms and conditions of using Antivir, and again if it is a product you like and find useful I’d recommend spending the money and purchasing it legally.
I’ll reproduce the information here from the wiki for ease of access and future referencing.
Windows XP Pro
1. Go to Start > Run.
2. Type gpedit.msc and click OK.
3. Navigate through User Configuration > Administrative Templates > System.
4. Double click "Don't run specified Windows applications".
5. Enable it and click show.
6. Add "avnotify.exe".
7. Click OK on all open windows.
Windows XP Home
1. Boot into Safe Mode (Press F8 while your system is booting up)
2. Log in as administrator.
3. Open C:\Program Files\Avira\AntiVir PersonalEdition Classic.
4. Right click on avnotify.exe and go to Properties > Security > Advanced.
5. Click on Edit-> Traverse Folder / Execute File-> deny-> OK.
6. Repeat for all users.
7. Reboot your computer normally.
Windows 7 or Windows Vista Business/Ultimate
1. Open the control panel through Start > Control Panel.
2. Go to Administrative Tools > Local security policy.
3. Click on Software Restriction Policy > Action > Create new restriction policy.
4. Right-click, and go to additional rules > new path rule.
5. Click Browse and navigate to C:\Program Files\Avira\AntiVir Desktop\ and double-click avnotify.exe.
6. Set the security level to Disallowed.
7. Click apply and OK.
The wiki gives details on removing the popup from Vista home so if you are running Vista home please hop on over to the wiki and check it out. I’m not yet able to confirm if these steps work but will no doubt check it out at some point in the future.