Inner tubing down rivers in the United States

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:

• Large rivers are ugly. They were the main transportation route for the first 200 years of this country's existence (including colonial era), and as a result became the spot for industry. Rivers are still the prime transportation for bulk goods, so be prepared for oil refineries, power plants, and other factories. Oh, and sewage treatment plants; there are lots of those.
• A tube doesn't travel very fast. It's meant for floating down a slow river on a summer afternoon drinking a beer. The shape presents about the maximum amount of resistance possible, and paddling with your hands is incredibly inefficient. I'd be surprised if you get over 5k a day -- about enough distance to get halfway from one town to another.
• There's no carrying capacity. OK, I suppose you can tow a drybag, but that just adds more resistance.
• If you get into an active shipping lane, you will die. A bunch of small boats get run over every year because their pilots don't realize just how fast a barge or cargo ship is moving.

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:

Pennsylvania has no general law prohibiting the launching of inner tubes or tubing on Pennsylvania rivers. [...] it is unlawful to launch or retrieve swimming aids, such as inner tubes and similar devices, from access areas managed for fishing and boating by the PFBC

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.