It is not illegal, but it is silly.
A DV, even if timely, imposes no requirement for any response on the part of a debt collector. A timely DV (i.e., sent within 30 days after dunning notice) imposes a cease collection bar on the debt collector, which then remains in effect until the debt collector chooses to respond. The debt collector can choose never to send validation, and simply accept the cease collection bar. With the debt being paid, they no longer have any debt upon which to conduct collection activities, so could care less about any bar on further attempts to collect on the debt.
They could simply deposit the DV in the trash.
It is more likely that a DV sent after payment of the debt would be well beyond the 30-day period for sending a timely DV, and thus would not even impose a cease collection bar.
Finally, sending a DV is based, as stated in FDCPA 809(b), on a consumer's dispute of the debt, in whole or in part.
It is thus improper for a consumer to dispute the validity of a debt that no longer exists.
It is a silly theory with no basis in law.
"FDCPA (b) Disputed debts
If the consumer notifies the debt collector in writing within the thirty-day period described in subsection (a) of this section that the debt, or any portion thereof, is disputed, or that the consumer requests the name and address of the original creditor, the debt collector shall cease collection of the debt, or any disputed portion thereof, until the debt collector obtains verification of the debt or a copy of a judgment, or the name and address of the original creditor, and a copy of such verification or judgment, or name and address of the original creditor, is mailed to the consumer by the debt collector. Collection activities and communications that do not otherwise violate this subchapter may continue during the 30-day period referred to in subsection (a) unless the consumer has notified the debt collector in writing that the debt, or any portion of the debt, is disputed or that the consumer requests the name and address of the original creditor. Any collection activities and communication during the 30-day period may not overshadow or be inconsistent with the disclosure of the consumer’s right to dispute the debt or request the name and address of the original creditor."