Is it possible to Trigger an exe to run on a failed build? Can you do this within Team City?
If you specifically want the failed builds, you can set up the dependent build as Eric said, and have that secondary buildscript use the REST API to pull up a list of the failed builds for the actual project.
If the latest build is in that failed builds list, then tell the build script to run the executable. If not, then you're all done!
http://confluence.jetbrains.net/display/TW/REST+API+Plugin