Dan,
If you are going to disk most of the cover in before doing the burn down, you will have most of the weeds seeds in the soil. A burn down will help more in the control of perennial plants and not much, if any, with annuals. The residue you plan to burn is better in the soil, work it in to help improve the soil organic matter. As for the weed problem, identify the weeds as they emerge and then apply an herbicide to control the weeds or a hoe.
Bill