My DS didn't have a coach fired, but twice his team switched coaches mid-season. The first time around, his coach also played for the USL club and ended up switching USL teams mid-season so new coach stepped in.
Second case, his coach also was an assistant at a local college and the college coaching duties took priority and he stepped down. Club also came through.
It really didn't impact things in a overally negative way.
Now, I imagine if the coach was fired for some sort of issue, that could have caused bigger issues.