What happened to his papers?

Did Leeuwenhoek leave any unpublished letters?

Yes!

Unfortunately, his papers are lost. His November 17, 1721, will states:

Ook is ons begeerte dat men in een kist ofte koffer sal opsluyten alle de ongedrukte schriften ende brieven, bij mij Leeuwenhoek sijn geschreven, rakende mijne ontdekkinge, nevens tien gesnede koopere plaaten, behoorende tot eenige ongedrukte brieven, waarvoor men seer na vijfhondert gulden heeft betaelt alsmede de vertaling in 't Latijn, daar men een hondert ende seventig guldens voor betaalt heeft ende dat soo lange als den schelm (dit woord is later door Leeuwenhoek geschrapt) Adriaan Beman in 't leven is, ende en sullen die brieven ende die naderhand nog geschreven sijn, ook niet mogen gedrukt werden bij desselfs soon ofte nabestaande.

It is our desire that in one box or suitcase will be closed up all the unpublished writings and letters, written by me, Leeuwenhoek, concerning my discoveries, and ten cut copper plates, belonging to some unpublished letters, for which was paid more than five hundred guilders as well as the translation into Latin, for which was paid seventy guilders and that so long as the villain [word deleted later by Leeuwenhoek] Adriaan Beman is living, both those letters and those still to be written afterwards, may not be printed by his son or next of kin.

From 1693 to 1702, Leeuwenhoek had published half a dozen volumes of letters printed by his next door neighbor Henrik van Krooneveld. For his Send-Brieven in 1718, he used Adriaan Beman. That relationship apparently did not go well and he died with another volume almost ready for publication.

Maria, as it turned out, survived her father by twenty years. Her will followed the spirit of her father's, though she had to make changes because several of the earlier beneficiaries had died. As he had instructed, the executors of her estate sold his microscopes "in a bundle" at auction. The Catalogus (left; click to enlarge) listed over two hundred lots of microsopes and their storage cabinets. Pre-auction, it was distributed via booksellers and auction houses in several Dutch cities in additiont to Delft.

The last page of the Catalogus (below right; click to enlarge) noted that the auction brought in a little more than 730 guilders. It had a nota bene concerning the papers that Leeuwenhoek left behind.

N.B. In den Boedel van wylen Juffr. Maria van Leeuwenhoek zyn gevonden eenige nagelate Manuscripten of Brieven van haar Vader, den Heer Anth: van Leeuwenhoek, dewelke door zyn Ed: in deszelfs Leven geschreven en in eene nette en goede ordre geschikt zyn, om als een vervolg op zyne voorgaande uitgegeve Brieven gedrukt te konnen werden; Alle de Platen daar toe behoorende, zyn daarby, en reeds in 't koper gegraveert, zoo als ook de Latynsche Vertaling van voorz. Brieven. Iemand genegen zynde, dit Werk te kopen, om het als een vervolg op zyne reeds uitgevene Brieven te laten drukken, kan zich addresseeren aan de Executeurs van de voorsz. Boedel.

N. B. In the estate of the late Miss Maria van Leeuwenhoek were found some manuscripts or letters left behind by her father, Mr. Anth. van Leeuwenhoek, which were written by him in his life and are arranged in neat and good order, to be able to print as a continuation of his previously published letters. All the plates belonging to these, are with them, and are already engraved in copper, as also the Latin translations of these letters. Someone being favorably disposed to buy this work, in order to have printed a continuation of his already published letters, can address himself to the executors of this estate.

Apparently, no one was interested because there is no record that anyone bought this lot at the auction. Nor were any posthumous letters published, so the box of unprinted letters must have been thrown away, perhaps burned, and the copper plates re-used for another purpose.

Which letters?

Were they the batch of four dozen between the last letter in Sevende Vervolg in 1702 and the first letter in Send-Brieven in 1712? Because Leeuwenhoek published only 40 letters in both Philosophical Transactions and his own volumes, these unpublished letters in Maria's estate could have been letters recording new observations. However, at least some of them would have been sent to the people they were addressed to. No such letters have surfaced, so perhaps the 1702-1712 letters are the better bet. In that case, we lost very little because all the manuscripts are still available at the Royal Society.