Er, this doesn't give much of a reason for how it avoids taxes.  The only argument I can discern from the graphic is that it's not taxable because the bitcoins are produced "without the involvement of government or banks".

Um ... yeah.  Good luck using that argument with an IRS agent.

(Note: this is separate from the issue of whether you can conceal your transactions from the IRS with bitcoin, keeping the government from ever finding out about this barter/private currency/whatever income.  But that's tax evasion, which is illegal.  That doesn't mean the TurboTax graphic's argument is remotely plausible if you ever do have to justify why you didn't list it as taxable income.)