The IRS has a section of their website, The Truth About Frivolous Tax Arguments dedicated to explaining the flaws of popular incorrect arguments of why taxes don't have to be paid.
Specifically, it says:
The requirement to pay taxes is not voluntary. Section 1 of the Internal Revenue Code clearly imposes a tax on the taxable income of individuals, estates, and trusts, as determined by the tables set forth in that section. [...]
Furthermore, the obligation to pay tax is described in section 6151, which requires taxpayers to submit payment with their tax returns. Failure to pay taxes could subject the non-complying individual to criminal penalties, including fines and imprisonment, as well as civil penalties.
They go on citing a number of case laws supporting this point.
Snopes agrees with this:
[I]n a legal sense, neither the obligation to file of tax returns nor to pay taxes owed is voluntary — those requirements are specifically spelled out in Title 26 of the U.S. Code, particularly Section 6151
The NY Times also agrees in their review of the film:
Arguments made in court that the income tax is invalid are so baseless that Congress has authorized fines of $25,000 for anyone who makes them.
The article goes into some more detail on the law, as well as the consequences of violating it.
For more details see also As a US citizen, what law requires me to pay income tax? at law.SE.
Regarding the movie itself, it might not be the best source. To quote Wikipedia:
The film has been criticized for its promotion of conspiracy theories, its copious factual errors, and its repeated misrepresentations of the individuals and views it purports to criticize.
