What questions should I ask as a candidate during an interview for a tenure-track assistant professor position? I will soon be at an on-campus interview
for a tenure-track assistant professor position.
I expect that in the course of the day,
I will be asked the question,
"Do you have any questions for us?"
What are good questions for me to ask
to show that I am interested in the department
and would very much like to join?
I don't think that most candidates are very good at faking interest. So, ask the questions you are genuinely interested in, and your interest will show through. It is true that you have to give the best impression you can during an interview, but you don't want to come across as fake or scripted. 

I would start by reading the university, college, and department websites. That may give you some questions. 

If you run out of questions, you can always ask other faculty in the department about themselves. It is rumored that academics love to talk about themselves.  You can also talk about your research, which is an easy way to make small talk and show off your communication skills. 

Given that general advice, here are some questions that I think are common and "safe":


Ask about the tenure process. How long is the probationary period? What is the general process? Is there any form of mid-tenure review? 
Ask about teaching. What courses do new faculty typically teach?  How tightly controlled are they? Will you have a lot of new preparations? Do faculty often teach over the summer? What process is used to make teaching assignments (seniority, rotation, etc.)? 
Ask about benefits. They can tell you about the insurance and retirement benefits, which vary significantly from school to school. 
Ask about housing. Where do most of the faculty live? How far from campus do they live? How expensive is it to live in the area? Are there any particularly nice locations to live near the college? 
Ask about living in the area. Is it a lively place? A quiet place that's "good for raising a family"? Is there much nightlife? Is there convenient local shopping?  Is it good for biking? 


Here are some questions that you have to think about carefully before you ask them:


Salary. Some prefer to talk about this during the interview; some prefer to wait until an offer is made. 
Two-body problems (if your spouse is also looking for a job at the university or in the area). Again, some prefer to wait until an offer is made, others mention it early. See these questions: Should one disclose his/her family information in an academic job interview? and When during the application process should a candidate mention that their spouse is also looking for a job?
Unusual goals. For example, if you are being interviewed by department X, but you want to work mostly with people in department Y. In some cases, this sort of thing is a mark in your favor - in others it will doom your chances of getting hired.