Can you be more specific? Is it all the audio that is missing (completely silent) or is it only certain aspects (voices, music, etc)?

I would suggest trying this:

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.