Update:
I found this in the error.log:
20-11-2008 11:56:40 1227178600 bank 1428 AAVM - scanning error: x_AavmCheckFileDirectEx: avfilesScanReal of C:\DOCUMENTS AND SETTINGS\ADMINISTRATOR\APPLICATION DATA\MOZILLA\FIREFOX\PROFILES\MZPVQBMS.DEFAULT\PREFS.JS failed, 00000005.
21-11-2008 07:26:38 1227248798 bank 1184 AAVM - scanning error: x_AavmCheckFileDirectEx: avfilesScanReal of C:\DOCUMENTS AND SETTINGS\ADMINISTRATOR\APPLICATION DATA\MOZILLA\FIREFOX\PROFILES\MZPVQBMS.DEFAULT\PREFS.JS failed, 00000005.
This shows that Avast tries to scan a file in the administrator account, when I'm logged on the "bank" account! This should give an access violation, of course. But, as I said, it must be an on-access scan. I don't run any scans when i do banking
I just run Firefox and Java.
The problem seems to be reproducible. Maybe it is related to logon. I have seen other files in DOCUMENTS AND SETTINGS being involved. Seems to be a little random.
Is there any way to find out what causes Avast to scan these files, like which program tried to access the file?