In plain English
FAR 52.213-3, Notice to Supplier, is a contract clause prescribed at FAR 13.302-5(c), most recently dated Apr 1984. 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 13.302-5(c), insert the following clause:
Prescribing reference: FAR 13.302-5(c).
The official text, verbatim
As prescribed in 13.302-5(c), insert the following clause:
Notice to Supplier (Apr 1984)
This is a firm order ONLY if your price does not exceed the maximum line item or total price in the Schedule. Submit invoices to the Contracting Officer. If you cannot perform in exact accordance with this order, Withhold Performance, and notify the Contracting Officer immediately, giving your quotation.
(End of clause)