Does driving an automatic saves diesel when using overdrive?

The lower the engine revolutions the least amount of fuel used. This applies to both diesel and gasoline engines. As long as you are not labouring the engine you won’t do it any damage. So yes, use overdrive when you can.

My 2000 Toyota Solara has a 5th gear overdrive and its responsible for the 31 mpg i received out on the interstate.

Use overdrive. like Darkscribe said, “the fewer the rpms, the better the gas or diesel mileage”.

Highly dependent on the vehicle.

For most vehicle it does, but I’ve had a few (like, half the cars I’ve owned) where the gearing and the power curve of the engine were/are such that it doesn’t. For instance, my current car only has ~70HP so if I hit any sort of incline in overdrive or exceed ~50 MPH then I have to put my foot into it enough that I use more gas than I would if I just kept my foot were it was, disengaged the O/D, and allowed the RPMs to rise from 2200 to ~3500.

Diesels are biased more towards low-RPM torque production, and most modern cars have enough power that they don’t labor at highway speed on level ground, but it’s literally true that your mileage may vary.