"We drove past that place" is correct. You may also say, "We went past that place." To use "We are past that place" you would have to be giving someone directions to where you are now. For example, "Turn right at the traffic light. We are past the Chinese restaurant." There's another connotation of "We are past that place." When I say, "I'm past [something]" it's like saying "I'm over it." It's an idiom that means that I'm so tired of this thing that I don't want to think about it anymore. For example, "I am so past taking exams. I've been doing nothing else all week." It's less common to refer to a place, but it's possible. We can say, "I'm so past (eating at) McDonald's. I think I'm going to puke if I have another hamburger."

These sentences use different tenses. 'drove' is past tense and 'are' is present tense. It depends on the context of who you are speaking to.

No, they are not interchangeable. "We drove past that place" could refer to 5 minutes ago, yesterday or long ago. Example: "Last week I drove past that new restaurant." "We are past that place" -- at least without more context -- implies that you recently passed that place during the same trip/journey/voyage that you are currently on. Example: "Mom, can we stop at the McDonalds? Sorry, sweetie, we are past that."