Back when these bikes were built Big Dog lined the tanks with that green liner you are seeing. That was before the government decided they needed to add ethanol to the chemical concoction we now call gasoline. Unfortunately the green tank liner material Big Dog used would not stand up to the ethanol in the new gas. So as you see it peels off. It is what it is. There are several threads on the forum about this issue. Some owners get in there and pull the loose stuff out and keep riding. Some owners strip the old liner out and do not reline their tanks and some go all the way and strip the old liner out and reline the tanks.
Don't forget to clean the screen on you're fuel shut off valve and if you don't have one yet you should install an inline fuel filter to keep that crap out of you're carb. Main thing is don't give up on you're bike. When you work through a few issues on it they are great bikes to own and ride.