The avast user of azureus has two choices:
1) the less desirable - exclude (the controlling process for azureus) javaw.exe in the [MailScanner] section of the avast4.ini file.
To me, this is the preferred option. Why not exclude javaw.exe and azureus.exe from MailScanner? Scanning BitTorrent protocol traffic with a POP3/NNTP scanner is pointless anyway.
2) the preferred option.
Establish yourself as an intermediate or advanced user in the Tools > Configuration wizard of azureus
In the Tools > Transfer window of azureus enter into the box for "Ignore peers with these data ports":
25;110
at least and preferably
25;110;119;143
to avoid conflict with the avast Mail Scanner.
This, to me, is less desirable, since it prevents connections to peers with stupidly-chosen port numbers. Yes, those peers shouldn't be using ports like 25 or 110, but sometimes you're shooting yourself in the foot by disallowing connections like this.