audio signal reconstruction from MFCC
Well, if the goal is to learn that some recording was similar to some reference, e.g. that the recording sound like the word "waffle", then yes, there's a way to reconstruct things, and that would be speech synthesis of "waffle", for example.

That's not a great example, but it's meant to illustrate one thing:

If your reduced version of the original signal, i.e. your feature vector (MFCCs in your case) is good enough to tell different things apart, and you can restrict the content of the signal to things that are actually well-represented by that feature set (e.g. you say "only voice, not music"), then yes, you can basically use closest-matches from a reference signal database.