First of all, how was the game installer? Which script did you use?
Before trying the follow steps, try using DXVK 0.81 and report if audio stays the same.
Probably unrelated, but you have to go with TKG wine to have DXVK 0.90 work properly (apparently).
Go on Lutris>Manage Runners scroll down to wine manage versions, download tkg-3.19
It contains a patched staging+esync+etc version of wine. Have that be your main warframe version. Go to your Warframe folder, there should be a launcher.sh or updater.sh file. Open it up, search for WINE (ctrl+f or whatever way). There should be a WINE/WINE_CMD or whatever environment variable (as well as a WINE64 variable). Change it to the one you just downloaded. For example, it should be similar to:
WINE_CMD = /home/YOURUSERNAME/./local/share/lutris/runners/wine/tkg-3.19-x86_64/bin/wine
Search for WARFRAME_EXE variable and make sure it’s warframe.exe, not warframe.x64.exe. The 64-bit version has a few problems with sound, afaik.
If this still doesn’t solve the problem, enable all sorts of debugging on Lutris and start it with
then see if you can find any sound error.
Another possible solution is to install the xact winetrick in your Warframe prefix and adding xaudio2_7(native,builtin) to the dll override list. This step should be made when you use the given install script.