Does the sentence mean that I feel very sad and because of that we can't talk tonight or does it mean that I feel very sad because we can't talk tonight? Or can it mean both?
It can mean both. Which it means will be determined by the discourse context. For instance:
I know we were supposed to meet for dinner to discuss the Anderson report, but my favorite uncle John just died and I feel so sad that we can't talk tonight.
Congratulations on your invitation! I know taking part in this conference will mean a great deal to your career, and you really have to attend. But I feel so sad that we can't talk tonight; I was really looking forward to seeing you. Have fun!
Context is a very large part of language; really, nothing means anything without context.
It can mean both, but it would be better to write 'that I can't talk to you tonight', if you want to say that you are sad. And if u mean the other meaning, avoid using 'so'.