Does child support have to be paid if the child is no longer living with the other parent?

Just to elaborate from a California law perspective: your order continues until a new order is entered, so if you don't want your ex collecting support for a child she is no longer supporting, you need to file an application to modify support right away.  Based on the facts you outline, you also may have a case for you to get custody and her to pay you support.  You should consult an attorney about all the ramifications of these developments.

You are obligated to pay child support either until the child is emancipated (not the case here, yet), or a court order ends that obligation.

So, you're going to need to apply to the court for this.  Under the law, the reduction or elimination of your child support can't be effective before the date you file that application.

There are a number of things that could factor into this.  I would not expect to have the support obligation completely eliminated, unless you decide you want custody and get that;  how much support you'll have to pay in the future, and to whom, I can't say.  But the law is that your child deserves your financial support, one way or another.