How should I dispose of material waste during home construction?
In my part of the world hiring a refuse service is standard and almost unavoidable. We had 30 cubic yard dumpsters (steel containers) on all our jobsites which were often emptied or replaced multiple times during the course of construction. Your contractor should have handled that. (If you are your contractor, now you know. :) )
They aren't cheap, so pack carefully and within the rules. Separate anything which can be repurposed or recycled to help save this ailing planet (and costs).