In plain English
FAR 52.222-16, Approval of Wage Rates, is a contract clause prescribed at FAR 22.407(b), most recently dated May 2014. The complete official text is reproduced below, verbatim, from GSA's published FAR source files.
Its text contains no sentence requiring insertion into subcontracts; see the flowdown section below for what that does and does not mean.
Does it flow down to subcontracts?
No flowdown mandate found in the clause text
We scanned the full clause text and found no sentence directing the contractor to insert this clause into subcontracts. That is a statement about the text, not legal advice: a prime contractor may still flow terms down contractually, an agency supplement (DFARS, VAAR, …) may add requirements, and clauses listed below (if any) may order this clause into subcontracts from the outside.
Where it's prescribed
As prescribed in 22.407(b), insert the following clause:
Prescribing reference: FAR 22.407(b).
The official text, verbatim
As prescribed in 22.407(b), insert the following clause:
Approval of Wage Rates (May 2014)
All straight time wage rates, and overtime rates based thereon, for laborers and mechanics engaged in work under this contract must be submitted for approval in writing by the head of the contracting activity or a representative expressly designated for this purpose, if the straight time wages exceed the rates for corresponding classifications contained in the applicable Construction Wage Rate Requirements minimum wage determination included in the contract. Any amount paid by the Contractor to any laborer or mechanic in excess of the agency approved wage rate shall be at the expense of the Contractor and shall not be reimbursed by the Government. If the Government refuses to authorize the use of the overtime, the Contractor is not released from the obligation to pay employees at the required overtime rates for any overtime actually worked.
(End of clause)