The opposing counsel was ordered to prepare Points and Authorities, but failed to do so, and now I want to ask for a dismissal of the request for order for failure to comply with the court's orders. What is my authority for this.

I don't know, because I don't know what/which Court and where you are. This is an International site.

It appears that OC filed a motion and the judge requested that counsel file a memorandum of the points and authorities to support his or her motion. In other words, I'm thinking that the court is still considering the motion.

Simply file something with the court pointing out that OC has failed to file the memorandum as instructed and therefore the relief requested should be denied. Submit an order denying the relief at the same time. Some places require a hearing before anything will happen. If that's the case in your jurisdiction, ask for a hearing.

In US courts, I would think the next step would be, if he doesn't meet the terms of the order, to file for contempt of court. The judge will probably warn them the first time and get things moving.

To answer OC's question, the "authority" for dismissing the request is the fact that OC failed to follow the court's request. That should be sufficient. If in doubt, cite the particular rule for the court which may require that motions, applications, etc. must be supported by memoranda of points and authorities (if indeed such rule exists in the particular jurisdiction).