I'm at a SLAC, so some of these suggestions might be slightly SLAC-centric, but most should generalize regardless of institution type.
- Collaboration: Nature/ease of collaborations within department, across departments, with faculty from nearby institutions (if relevant)?
- Research labs: What do people's labs look like (literally and figuratively)? Number of students involved, roles and duties w/in lab? Do undergrads get paid for working in labs? Course credit? Do they just volunteer?
- Tenure requirements: What's expected for receiving tenure (with respect to research, teaching, and service)? Pre-tenure review(s) (how often, what do they entail)?
- Courses: How are courses assigned to faculty members? What does a typical first-year faculty member's set of courses look like?
- Grants: Expectations for writing (and receiving) grants?
- Service: Service requirements/expectations for junior faculty (committees, advising, etc.)?
- Departmental/institutional access to various resources relevant for your work (stats software, experimental presentation software, online survey access, etc.)?
- Internal funding: Funding available for paying student researchers over the summer? For conferences, membership dues, etc.? (Or do such funds come out of your startup and/or grants?)
- Subject pool: How large is it, how is it managed?
- What do they like about living in the area? What are the opportunities for doing X (your hobbies, interests)?
Don't be reluctant to ask the same questions to multiple people. It's often useful (and reassuring) to hear convergent answers and opinions; but divergent ones can be similarly informative.