You don't have copyright to this material. Presumably, somebody that's not you does. That somebody is either the person that wrote the material, or perhaps the school that the person works for. There are cases to be made for both.
In any case, if you decide to share the material with Professor B, you would be doing through one of two pathways: either in receiving the material, you have been given some form of license to share it, or you would be sharing it claiming "fair use". A third possibility is that the material was written and already given some sort of license (similar to a stack exchange post!), and the conditions for sharing are already clear. I'll ignore this scenario, as you would probably know about it if it were relevant here.
Let's talk about any implied licenses you may have been given that would allow you to share it. If this is the case, you'd likely find reference to it in your school's policy materials or your course materials. At my own school, unauthorized sharing of course material is specifically listed as a violation in our Academic Honesty Policy. In my specific coursework, in my syllabi, I state that sharing is not authorized. I don't know what prof A's policy is, but you can certainly look in the syllabus and/or course site to try to figure it out.
"Fair use" is a legal term, and whether fair use applies to the scenario you describe is arguable either way. That's how these things work. There are those that will say "Fair Use applies", and those that will say it doesn't. If you would like to figure it out, a librarian would be specifically trained in this area, and would be able to help you. University librarians seem particularly knowledgeable in this area, because these issues come up all the time as profs prepare course material.
Beyond a librarian, there are four factors that are used to figure out if fair use applies. The exercise is often unsatisfactory, as the answers are often not clear.
This library (https://library.madonna.edu/copyright/fairuse) offers some good guidelines. This figure that they provide illustrates all of them. I particularly like this figure, because of how clearly it shows that fair use is a continuum, and not a black and white binary issue.
The same library also offers a flow chart of how the determination is made:
The big point in the flow chart, that often gets unfortunately ignored, is that you need to investigate all four areas to make a strong determination of fair use -- one of them is not enough.
In your specific case, the fact that you'd like to take essentially all the course materials and hand them to someone else seems to violate criteria two, pushing this away from fair use in the continuum. I'd suggest that criteria 1 is also violated, because prof B may end up using the material in exactly the way it was meant to be used. Also, if the material is out there, it may become harder for prof A to make a living by teaching the stuff.
Further, the original material may be bound by some license. For example, adopting a textbook often confers upon the instructor the right to use art from that book in their teaching materials (when properly acknowledged). In this case, sharing those materials with prof B might violate prof A's de facto agreement with that publisher.
Lastly, when you're sharing stuff that isn't yours, the polite thing to do is to ask the person who's stuff it is if they're OK with that. It's simply the mannerly thing to do. "The prof is unresponsive to email" simply isn't a good enough reason to ignore this.
