They mean different things.
We live in a big house with a white door.
This is a relatively standard description of the house one lives in, but the listener is not meant to actually know what house you are referring to.
We live in the big house with a white door.
This is a relatively strange description because of the mismatch "the"/"a". You are indicating that you expect the listener to know what house you are referring to, but not which door. This suggests, for example, that the door is interior to the house and is of some sort of importance to the listener, rather than being an arbitrary identifying characteristic:
"No," Sarah said, "You don't understand. The door is broken and my parents are going to kill me if they get home and see that I threw this big party."
My friend Josh spoke first. "I mean, if all you need is a door, we could probably just swap your door with the one from my room. I'll just tell my parents that I accidentally broke it."
Sarah declined. "Josh, I've been to your house, it's halfway across town. It'd take too long. Plus, your door is brown. My door is white."
"We could paint it," Josh offered.
"With what paint?! And how would it be dry by tonight?!" she fumed.
I looked to my sister and she nodded. I stepped in. "We live in the big house with a white door. I don't know how I'd explain that it broke to my folks, but I bet we can come up with something."
"The one just down the way?" Sarah asked.
"Yep, that one."
"Let me see this door."
Most likely you would instead see a completely different statement,
We live in the big house with the white door.
This is a relatively standard identification of which house belongs to you, out of a collection of other houses where some of them have white doors but one in particular stands out as "big".