I've got this crazy idea to have an adventure trip inner tubing down a large river in the USA and stopping off at various locations along the way. Unfortunately, I can't seem to find any information as to whether or not this is legal. In general, what is the policy about inner tubing down rivers? Where would be the best place to find out if it is allowed in a local river?
I'll start by saying that you're right, the idea is crazy. However, your idea of a "large" river may be different from mine. I think of "large" as anything over 100 meters across that has commercial traffic: the Mississippi, to be sure, but also the Hudson, or even the lower Delaware. Here are the reasons I wouldn't do it:
My recommendation, if you want a river adventure, is to use a sea kayak. It doesn't have anywhere near the Huck Finn gestalt, but is a lot more practical.
I'd also recommend staying off large rivers. Aside from the negatives listed above, in my experience smaller rivers are much more interesting. Even man-made rivers: I've been thinking about a canoe/kayak trip on the Erie Canal.
But, you asked about laws. As one of the commenters said, it will depend on local state laws. You can generally find those laws by Googling for "STATE fish and boat" (generally fishing and boating are controlled by the same agency), or "STATE unpowered boat" (which might or might not take you to a page showing the rules).
In Pennsylvania (where I live), unpowered boats under a certain length do not need to be registered or licensed, unless you want to use a state-managed launch (although enforcement is rare; I've only seen rangers at a launch once). The rules do specifically address tubing:
So, if you want to float past the refineries of Philadelphia, you can do it legally. Although you might have to take a copy of the rules to show to the police.