Short answer: You should not have an avatar with copyrighted content.
Long answer
According to the Stack Exchange TOSStack Exchange TOS all Subscriber Content must be irrevocably licensed to Stack Exchange under the Creative Commons Attribution Share Alike license. This requires us as members to have the right to give them such a license.
If a copyright issue occurs, normally the real copyright holder will send a DMCA takedown notice (an e-mail) to Stack Exchange and they presumably will remove your avatar. You are the one who originally pirated the image so you are still personally responsible and can be held accountable for the violation. Stack Exchange is probably also responsible, but only so far as neglect goes (which is not that far if they act upon all requests from rights holders), but they can then in turn hold you as a member responsible for paying their fees arising from your copyright violation.
The TOS even covers this in detail:
Subscriber represents, warrants and agrees that it will not contribute any Subscriber Content that (a) infringes, violates or otherwise interferes with any copyright or trademark of another party
...
Subscriber will indemnify and hold Stack Exchange, ..., harmless, including costs and attorneys' fees, from any claim or demand made by any third party due to or arising out of Subscriber’s access to the Network, use of the Services, the violation of this Agreement by Subscriber, or the infringement by Subscriber, or any third party using the Subscriber's account, of any intellectual property or other right of any person or entity.