For me, the scariest part was the whole game. I played the Original RE1 on PlayStation; I had Resident Evil: Director's Cut.
I don't know about anybody else, but I only ran into one other person who had the same problem. I bought a cheap ass controller from K-Mart that somehow shorted my PS' board so Memory Cards were completely Non-Functional.
My first ever play ever Resident Evil was with no saves whilst a buddy watched. It was sheer tension the entire way through, and yes I did beat it. I also did the same thing for Resident Evil 2 several times.
However, Resident Evil 3 made it a little more difficult with cheap deaths, so I never actually beat that one.
Anyway, the scariest part of Resident Evil for me was the hunters of course, because they would kill you instantly. For me, it was Game Over. The final countdown at the end with Tyrant was pretty damn nuts also, what a sense of accomplishment, I felt like I actually Surived a Horror.