Go to "My Documents -> Escape From Tarkov"
Right click on "shared.ini" and click "Properties"
In the pop up window, click "security" and then click "Authenticated Users" and then click "Edit"
Then click the "Full Access" box under "Allow" and then boom, your voice-lines will now save when you hot-key them in-game.
Edit: Disclaimer, this may not work for everybody. Worked for me and the few people I recommended it to. Make sure this file is NOT read only and I personally made sure every user listed had full access to the file. Hope it works for you guys like it does for me.