What would you do?

A legal analysis from an actual, practicing lawyer.

First, I assume we are in America. I can't speak to the laws of wherever it is that "Johnny" lives currently, likely somewhere over the pond.

Ok, so, first -- does Johnny have any responsibility to Tommy?

"Johnny doesn't mind this, as it basically means he has a furnished bedroom to offer to guests when they come by."
Johnny actually does have a responsibility under the law of bailments. See http://en.wikipedia.org/wiki/Bailment
This is a "mutual bailment" in that Tommy benefitted from being able to store his furniture and Johnny got to have a furnished room for guests. The responsibility that Johnny has to Tommy is that he must exercise ordinary, reasonable care to prevent the furniture from being harmed. In other words, Johnny cannot intentionally trash the stuff without Tommy's permission, or at the very least, without giving Tommy a reasonable opportunity to retrieve his items.

Second, should Billy have to pay Tommy?
"Tommy comes to visit, and hangs out with Billy, Billy's brother (who is also in town), and Johnny for a few days. While he's in town, he talks to Billy's brother about selling his furniture to Billy. The brother seems agreeable to the prospect of Billy buying the furniture, but the subject is never actually brought up with Billy. Tommy leaves thinking he has sold his old stuff."
Billy almost certainly should not have to pay Tommy. However, there are circumstances consistent with the above facts in which Billy could have to pay Tommy. First, if Billy authorized his brother to negotiate on his behalf (for example if Billy hired his brother as his attorney or agent or even his furniture consultant)--then if Billy's brother made a deal, Billy needs to pay. Second, if Billy's actions would reasonably lead Tommy to believe that Billy's brother had the authority to bind Billy, then Billy would have to pay. For example, if Tommy said, "Hey Billy, you interested in buying my furniture?" and Billy said, "Ask my brother, he knows furniture and handles my furniture deals." This is the law of agency. See http://en.wikipedia.org/wiki/Agency_(law)

Even if Billy's brother was authorized to make a deal that would bind Billy, do we have an enforceable contract?
That depends on a few things:
First, obviously there is no written, signed contract, or we would not have this problem. In the absence of a written contract, we have to prove an oral contract. The sale of goods (like furniture) is governed by article 2 of the Uniform Commercial Code. Under the UCC, a contract can be any offer and acceptance and can be proven by mere conduct that demonstrates that a deal has been reached. So, IF Billy's brother was authorized to deal for Billy, WHAT HE ACTUALLY SAID TO TOMMY is important in determining if there is an enforceable contract. UNLESS -- the sale price is over $500. Under the "Statute of Frauds" provision of the UCC, contracts for the sales of goods over $500 must be memorialized in writing. So, in order for Tommy to collect from Billy, he needs to be able to show 1) that Billy's brother had actual or apparent authority to make a deal on Billy's behalf, 2) that Tommy and Billy's brother actually reached a deal, and 3) that the deal was for less than $500. If Tommy was trying to hire me to take his case, I would tell him that it was a bad idea, but that I would be willing to represent him for $20,000.00 paid up front (this is how I handle refusing representation in bad cases -- because, who knows, maybe some fool will take me up on it some time).

Can Tommy leave his property at Johnny's house (in Billy's room, no less) indefinitely and forbid Billy from using the property?
No. Of course not. Assuming we have a reliable narrator, Tommy is being a fucking tool. First, when Billy moved in, Tommy's property stopped being beneficial to Johnny -- in legal terms, the bailment was no longer mutual, but rather gratuitous. When a bailment only benefits the property owner, the holder of the property only needs to refrain from GROSS NEGLIGENCE in preventing harm to the property. In fact, Johnny (or Billy) can likely do what they want with the property--trash it or keep it, if they give Tommy a reasonable opportunity to get his stuff. Tommy sounds like a baby. He needs to grow up. He also needs to come get his shit out of your pad.

I practice civil litigation (lawsuits not involving family law or criminal law) in Texas. PM me if you live in Texas need a good lawyer.