To make something go away means to eliminate something, to make something disappear or to resolve a problem.

1. My doctor said that the tablets she prescribed should make the inflammation go away within a few hours. 
2. The politician employed a PR firm in the hopes of making the scandal go away, but it was already too late.

1. Burying our heads in the sand is not going to make the problem go away! We need to tackle the issue head-on.
2. I don't care how we do it, we must make this scandal go away or it will ruin our reputation.