Instead of trying to use his lightsaber to cut through the doors when trying to get to the vicroy, why didn't Qui-Gon just use the force to unlock the doors and open them? Seems like that would be much easier and quicker.
That assumes the mechanism was simple enough for him to manipulate. The doors were security doors (also called blast doors), obviously intended to keep those on the bridge safe from attack. While they may not have been designed specifically with the Jedi in mind, there are other Force users in the galaxy. A complex locking mechanism that would physically block the doors from being moved out of place, along with some type of physical combination, would have easily made it difficult for a Force user to manipulate the doors.
Almost any kind of physical combination device would have required a Jedi to take the time to sense where everything was, as well as figuring out combinations that would work, and then being able to manipulate all the moving parts. Just sussing out the combination could have easily taken longer to do than cutting through with a lightsaber.
He was trying to get through a blast door on a warship, not a screen in a private house where you can probably Force-pull a hinge or move the piece in the simple lock.
The Force exerts a certain amount of force (no pun intended). It is likely significantly less than what is required to break through a durasteel blast resistant door.
It's as simple as that the more complex a lock is, just as a puzzle, it is harder to unlock by manipulating the pieces. It is just much easier to cut through them with a lightsaber. If it were a simple skeleton key lock or a door with tumblers, it would be fairly easy it instantly unlock it, but like a bank vault imagine multiple crossing steel bars and thick plating on top too. That's a lot of effort when you can just cut it open.