What material can be used as a substitute to plastic or paper for packing purposes?

That depends on what you're packing, and why you want to avoid paper and plastic.

One option to consider is textiles.

Pack fragile items such as dishes between layers of towels or clothing when moving from one home to another. The fabric serves as padding, and you don't need to pack it separately. (This does assume some hard-sided container. If you'd like to save the environment, you could always collect used boxes from a supermarket or from someone else who has recently moved, then be sure to pass them along or recycle them.)

I make gift bags out of fabric scraps and use them over and over instead of paper gift wrap. My usual rule is that if people love them and want to keep them for their own reuse, they may; otherwise, I'll collect them and reuse them for next year.

A blanket I got as a gift came with a strip of fabric wrapped around it. I stitched the sides together, leaving a Velcro closure where it was, and use the resulting bag to store and keep dust off a pair of shoes I only wear on rare, dressy occasions. (Of course, some paper and plastic packaging is designed for reuse. Many tools come in plastic cases that are designed for storage and transport.)

Another option is wood. Currently at the store near me, there is a kind of cheese packaged in six wedges (plastic wrapped), in a shallow cylindrical tray made of thin wood. There is also at least one wine where the glass bottles come in a wooden box. The wood is not lighter than a corresponding cardboard box would be. I suspect the choice is as much for style as for structure.