No, the free version of Ad-Aware does not include any protection running in the back ground that needs to be disabled and if TeaTimer isn't running you don't need to disable anything there either. I believe what is being suggested is to temporarily disable your firewall and antivirus when performing certain tasks.
You shouldn't have to do this when updating a program like Ad-Aware but sometimes it helps. However, when installing any kind of security program, and sometimes just any program, it is always a good idea to disable your firewall temporarily--if nothing else because some will drive you nuts warning you about all the files that are being added or changed. Most antivirus programs lately are including Intrusion Prevention Systems that work similar to firewalls and other protection measures that can also cause problems, so disabling them when installing will help with a clean and smooth install.
If you read the Lavasoft support thread
ClausValca posted, you'll see this is what is being suggested. Since that is posted at their forum, it should fix the problem you are experiencing and you won't have to change any settings on your PC. Have you tried that yet?
There are some anti-spyware apps that have real time protection that can interfere with changes that you want to make or pester you asking if you want to allow them. Mostly that happens with apps you pay for like Ad-Aware's Pro version that includes Ad-Watch. However two free ones that I know about that can also do this is Spybot's Teatimer and MS Antispyware.
I wouldn't be so worried about these type programs that just scan and clean--if you don't like that it has this problem and don't want to use their fix for it, use something else--there are plenty of good free antispyware programs available. It's just better to prevent infections in the first place rather than have all these cleaning scanners designed to help you after you get infected.
As far as safe mode there is a tutorial here:
Safe ModeThe reason to run removal scans in safe mode is because only the most basic drivers and programs are running in there--you will see far fewer processes in Task Manager there. In normal mode malware that may be running can have ways of protecting itself from removal. In safe mode it won't be running (except in some extreme cases), so detection and removal goes much more smoothly.