Version 4.1.06 of the TonePrint application stores user TonePrints in a file named "database.sqlite3" in the user's app data folder, which, in Windows 10, is a hidden folder named "C:\Users\[user name]\AppData\Roaming". If you want to replace your user TonePrints on one computer with user TonePrints from another computer, just copy this file from the one computer to the other.
If you want to copy TonePrints from one PC to another, but you want to keep your old TonePrints, too, simply copying the file won't work, because it will overwrite TonePrints. For that purpose, I wrote some scripts for Windows PC that will grab TonePrints from one PC and install them on another PC without deleting the TonePrints that are already there. You can download them in a zip file here:
To use them, download the zip file, extract the contents to an empty folder, then follow the instructions in the file named "README.txt" that is included in the zip file. They should work with version 4.1.06 of the PC TonePrint application.
I don't know how to transfer user TonePrints from a smartphone to a PC, or vice versa. The smartphone app might have a database.sqlite3, but I have not been able to locate it on my phone.
UPDATE: the scripts won't work for new versions of the TonePrint application (e.g. 4.4.03). The database file name apparently changed to "databaseV1.sqlite3" or something.