No.
However, if you set mutually exclusions between OA and Avast, OA's Banking Mode (which is not available in the Free Version though) will not work. OA's BITS protection should work either way, but I'm not sure about the undesired domains blocking.
Please note that is not an issue between Avast and OA exclusively:
If your anti-virus solution uses a proxy to implement its web or email scanning capabilities and you excluded your anti-virus software in Online Armor then the Banking Mode may have no affect.
(source: see the link below)
You can read more about the topic here: hxxp://support.emsisoft.com/topic/5208-known-online-armor-issues/page__view__findpost__p__31544
