People who say "we need a world without money" are actually saying "we need a world without trading and barter," for the two are the same.
Money is simply the good for which people most commonly barter. Remove any form of money from society and you'll quickly observe people finding other things to trade with.
If trade and exchange are occurring, then goods are being traded between people. Whichever good is most commonly used in trade is given the name of money. Not only is it a silly idea to "remove money" from society, but it is in fact as impossible as removing speech.
No. I think this is wrong and that it's important to say why it's wrong.
Forgoing the legal definitions of what fiat currency is, money, generally speaking, is something of value. And, as you said, it is the most commonly traded good.
But a world without money is NOT the same as a world without barter.
The thing about money is that it displaces value away from all other objects and redistributes it according to the value of the accepted currency. I'm a musician. I love instruments because I can play them; playing them is intrinsically beneficial to me. I'd much rather have my guitar than, say, a watch. But because money exists, if that watch happens to be a Rolex, you better believe I'd rather have the watch.
One of my first posts I made was about this same idea. I still believe it to be true, and, unfortunately, I believe BTC does nothing to escape it. In my opinion, an all-barter world is ideal, ignoring all of the psychological and other considerations that make it currently impossible (e.g. bye-bye global economics and mass production. "Yeah, we'll give you 10 million plastic spoons for 50,000 boxes of Captain Crunch."). Still, I think it's money's ultimate negative effect on society and people in general.