Very low current fuse (100 µA)
I can't directly help you with a method, but there are a number of medical "disposables" that time out and cannot be reused.  I believe attachments for the DaVinci surgical robot work this way.  You're having what we call an "XY problem".  You're not really looking for a fuse, you're looking for a way to identify a disposable that has already been used, presumably via electronic means, presumably low cost, and presumably with some realistic interface requirements that you haven't shared.  You'll get a much better answer if you spend the time speccing out your real problem than walking us into the middle of it.

A variety of approaches may work.  Anything that would provide a read-only unique ID on the bandaid would allow you to maintain a database on your host device, and you can then upload to some central database if necessary.    Microchip has a unique ID product at $0.17 USD in 5K quantity: http://www.microchip.com/wwwproducts/en/24AA02UID   -- I'm sure there are many competitors

In fact, any small writable memory device on the bandaid would allow you to set a "used" bit by some circumspect method.  This might even be more difficult to build a work around for than a "fuse" which can always be mechanically shorted.