Please double-check that all the volume sliders in the Audio Options menu are set to 100% on.
If that doesn't work then check your Windows sound settings to make sure you have everything hooked up correctly. Also, in Options->Sound check to make sure the audio output device is the correct one for your system.
If trying the above does not work then try this:
1. On the Warframe launcher, click the little gear icon on the top right. In the Download Cache section hit "Verify" and "Optimize".
2. If that doesn't work then please try re-installing the game.
Let us know if any of these solutions work.