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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.

No it is not true. The Sixteenth Amendment was adopted in 1913, so taxes not apportioned to states are constitutional. US Code 26 Subtitle A is the relevant law.

The quoted text seems to allude to a number of arguments that income tax does not need to be paid. The IRS has a page refuting frivolous tax arguments, and anyone contemplating trying to claim that they don't need to pay income tax should read it first. Jonathan Siegel, Professor of Law at George Washington University Law School, also has a similar collection of tax myths, including a page specifically dedicated to refuting the claims made in America: Freedom to Fascism.

Regarding the quoted text:

• "(in the private sector)": Presumably refers to the contention that only federal employees need pay federal income tax. This is based on a misreading of the law, which states that the term "employee" includes federal employees (my emphasis). Some people think that "includes" means "includes only".




• "Direct unapportioned tax": Presumably this refers to the contention that the Sixteenth Amendment was never properly ratified, or does not authorise federal income tax. No court has ever accepted this argument..




• "Labour or compensation, or services": Presumably this refers to contentions that wages are not "income" because they were received in direct exchange for a thing of equal value (i.e. some work). The courts have never accepted this argument either.

Let me take a different tack on this. The statement as presented,

There is no law which requires the average American worker (in the
private sector) to pay a direct unapportioned tax on their labor and
compensation, or services.

is exactly (in a larger sense) correct, but this larger sense does not help the speaker.

The 16th amendment specifically permits an unapportioned income tax.

The Congress shall have power to lay and collect taxes on incomes,
from whatever source derived, without apportionment among the several
States, and without regard to any census or enumeration.

The internal revenue code (US Code, Title 26) is enacted by Congress, imposes the various revenue taxes, and provides penalties for non-compliance. If this is not a law, I don't know what is.

Thing is, an income tax (as described in the OP) is not a direct tax, and hence does not need to be apportioned. The meaning of "direct" is often confused, and the fact that a taxpayer pays "directly" to the IRS is often the underlying source of confusion.

The Constitution provides that

No Capitation, or other direct, Tax shall be laid, unless in
Proportion to the Census or Enumeration herein before directed to be
taken. - Section 1, Article 9, Clause 4.

This refers to a head tax, but there are (in US law) only 3 kinds of direct taxes:

Only three taxes are definitely known to be direct: (1) a capitation [
. . . ], (2) a tax upon real property, and (3) a tax upon personal
property. - Murphy v. Internal Revenue Service and United StatesLink

Direct taxes are taxes "directly" on property. Indirect taxes are taxes on transactions, or transfers of wealth. And most income falls squarely in the latter category.

In fact the apportionment/enumeration requirement is so burdensome that the Federal government has never even bothered to try for a national property tax or real estate tax. In this day of computers and networked data bases it would probably be doable, but not in the past.

Of course, you might ask, why the confusion? Well, that goes to Pollock v. Farmers' Loan & Trust Company, an 1895 Supreme Court decision that held that a tax on income derived directly from property (such as rent or dividends) was in fact a direct tax.

...taxes on real estate being indisputably direct taxes, taxes on the
rents or income of real estate are equally direct taxes.
Citation

It made no mention of earned income (as from a job), and so does not apply to what most see as "income tax". The decision invalidated the first general income tax, the Wilson-Gorman Tariff Act of 1894, and promised to really complicate bookkeeping for any subsequent income tax. The 16th amendment was a direct response to this situation.

So, as I started out saying, the statement is precisely correct. There is no law which requires payment of a direct tax (unapportioned or otherwise) on labor, compensation or services. But. The IRS income tax on these categories is an indirect tax, and that tax must be paid. The fact that you will pay directly to the IRS does not make it a direct tax (within this context).

From the Wikipedia article on this film,

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.

More to the point, further on the same Wikipedia article notes

One of the listed stars of the film, Irwin Schiff, was sentenced on February 24, 2006 to 13 years and 7 months in prison for tax evasion and ordered to pay over $4.2 million in restitution.

So we have here a film in which claims are made that the Federal income tax is illegal. At least one of the people making these claims, Mr. Schiff, was sent to jail for not paying said tax. Thus, it would seem that either there is a huge conspiracy of governmental actors, including the legislative, executive, and judicial branches and their various minions, who are acting together in a coordinated manner to subvert the precious rights of Americans to not pay income or any other tax - or else this film is a load of hogwash.

Frankly, I'm going with "hogwash".