Can the word "over" be replaced by "due to" or "because of"?

Yes, you could say either

"I couldn't hear what he said over/above the noise of the traffic." or

"I couldn't hear what he said due to/because of the traffic."

Any of those expressions would be perfectly understandable.

The first two, "I could not hear him over/above the traffic," implies that in order for "him" to be heard, "he" would have to make his voice louder than the traffic--his voice would have to rise above the noise of the traffic--in order to be heard by "I".

The second two combinations simply mean that the traffic was too loud--and "he" was too quiet compared to the traffic--for "I" to hear what he was saying.

So, while there may be slight differences in exact meaning, you could say any of these, and no one would be more correct than the other.