There's a general awareness that smoking is harmful.
It would not sound very bad if you left out the word "that" but I think most people would consider it incorrect. The original sentence with the "that" is better.
I'm not an English teacher by any means, but I'm a native. I can give you my opinion about the sentence.
So, I think of the sentence: "There's a general awareness that smoking is bad," as essentially having two parts, and that the word "that" is used in the sentence to connect the two parts of the sentence together.
Part 1: There's a general awareness
Connecting word: that
Part 2: smoking is bad.
There's a general awareness that smoking is bad.