Occasionally when I have my unfulfillable inventory returned it is broken or damaged. Should I remove my unfulfillable inventory and have it sent back to me or should I have Amazon dispose of it? How do I decide which option to use?
We have everything sent back to us, as sometimes there is nothing wrong with the item and we can sell it again. Even the opened items that we can’t sell as new usually have some value elsewhere. Some of the items we sell, I have used the returned items. Especially boat items and kitchen appliances. 
It’s not an official check box on weather or not to sell, but “Would I want a return of this?” is a little tiny, subliminal check box.
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I get it sent back because it is generally in great condition 
Most of the time inventory that Amazon has marked as “damaged” is not actually damaged. But when it is truly damaged this can indicate a potential issue with product packaging. While normally it cannot be resold as New often it can still be perfectly fine for personal use, (re)gifting or depending on the category perhaps sold as Used. Also I like to personally review returned merchandise to see which items have been worn or used before being returned. Which is a lot more often than you might think, but just part of doing business.
I always have it sent back to me; for 50 cents why not. They usually come back on open box condition. I sell it on eBay, usually at my break even point. That way I still get my money back. In some cases I give stuff away if it’s not worth the time to sell it.