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I would advise to trim these during dormancy, but when the temperatures remain over 10C. This is, usually, right before it starts putting on new growth. This article will help:

https://www.gardeningknowhow.com/ornamental/trees/fruitless-mulberry/how-to-prune-a-mulberry-tree.htm

This is very common, and it is not a cause for concern unless you don't get any ripened fruit at all. This is when you know you have an issue. Otherwise, this is done because most of the fruit is lost to pests, anyway. Years of adaptation has led the tree to overbear, so that it has a better chance of ripening a few fruit for reproduction.

Now if you don't get any fruit at all, you will want to pinpoint whether it is pests or just not ripening at all.

This article will help: https://www.gardeningknowhow.com/ornamental/trees/fruitless-mulberry/mulberry-fruit-drop.htm

This may be a mild case of Cercospora leaf spot or the beginnings of another disease. Cercospora leaf spot causes brown spots on leaves which can also develop into small holes, and some leaves may fall off. It shouldn't hurt the tree unless the symptoms are severe.
https://plantdiseasehandbook.tamu.edu/landscaping/trees/mulberry/
http://www.extension.umn.edu/garden/yard-garden/trees-shrubs/management-of-leaf-spot-diseases/

I would not recommend burning the leaves, as this will completely destroy the good compounds within the leaves. I would air dry them before you drank them, rather than burn or cook them.

I found this discussion on growing mulberry as a shrub or weeping tree. https://www.houzz.com/discussions/1512692/how-do-you-grow-mulberries-as-a-tree-or-pruned-as-a-shrub
You are fighting mulberry's natural growth so it will likely need pruning twice a year. Pruning technic is crucial to avoid witches' brooms and water-sprouts.

If your goal is to kill the plant, girdling usually takes place closer to the ground. You will have new growth the entire 3 ft to the ground and probably some suckers as well. I suggest immediately applying glyphosate or triclopyr to the fresh stumps of new growth. It just takes a drop or two. I apply with a small foam paintbrush. To avoid herbicide use, you have to keep pruning new growth every few weeks - I'm guessing here - throughout summer and fall to exhaust the root system. Allowing leaves to stay on the plant starts to rebuild the roots and prolongs the process.