We intake floppies at our facility and send the bulk to a reseller in California. They scrub and refurb undamaged disks for a small but strong demand for them. They cover shipping at Media rates so as long as you have about 200 in the box and ship that way, it's free. They take ZIP and Jazz discs, too. Just remember that blank, unused discs can't go Media Mail.
If you find floppies of games, OS boots, and device drivers, you really ought to explore online sales. As people try to salvage older machines, they are finding the manufacturers and programmers have long since dumped these older software sets and secondhand copies often do not exist outside of physical media. I have had a lot of luck selling old DOS 3.1 start disks and flight sim games from the 1990's and a friend does a steady business in drivers for Mac machines. 5.25 disks without otherwise collectible content may, themselves, have value to vintage gaming enthusiasts who write to them. They were not as heavily over-produced as the 3.5's which followed.
As scrap, they are functionally without value, as are tapes. No one I've ever spoken to can extract any valuable metal content from tape or disk. I heard, once, that old oxide tape mylar had a small amount of silver. I've spoken to a couple chemical processors who were patient with my questions but ultimately concluded that the low value of silver, at present, provides no incentive to even bother with exploratory reclamation and that there is likely none to gather, anyhow.
Data backup tapes like LTO, Travan, DC, etc.. may have resale value with one of the various, online "SELL US YOUR TAPES" processors and I can offer references.
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