Today is an excellent example. I've pushed commits in the "feature-online-installer" branch. The Travis build has failed. So, in your branch (feature-travis), the readme shows a "Build failed" image.

If we keep this image as it, all readme in all branches will displays a "Build failed", even in Master or Develop.
But this doesn't mean the master branch is not safe...

Github have a good integration with Travis build. Every commit displays its build status, and every Pull Request shows information about the lasted build in the branch. Adding, or not, this image doesn't change the travis activation.
I think we must keep Travis integration (with the Github hooks), but not this image, it can display wrong status on safe branches.