I am interested in a tenure-track position at a university with 9 months hard money and 3 months soft money during the summer (by this I mean something between a liberal arts college and an R1 research institution). However, I'm concerned about my lack of teaching experience. I was a TA for some lab courses in grad school but that's about it. Now as a postdoc, would mentoring a summer undergraduate student help my application in the "teaching" qualifications that hiring committees look for?
If this is all you've got that go ahead and put it in your applications. However, you'll still look very inexperienced in teaching in comparison with applicants who've actually taught classes.
The question is about teaching experience and positions in the US at institutions that are somewhere between liberal arts colleges and R1 universities. The question doesn't mention the discipline, and this could be important.
I've chaired several tenure track search committees in mathematics over the last few years at such an institution (my institution is a "School of Mines".) We get hundreds of applications for every tenure track position and everyone we've hired recently has had significant teaching experience, typically including working as a TA during graduate school and then doing additional teaching of classes as part of postdoc or visiting assistant professor positions for one or more years.
In my experience, new tenure-track hires in other science and engineering disciplines tend to have somewhat less teaching experience than in mathematics. However, I'd say that most new hires at my institution have at least one or more semesters of experience teaching classes.
If this is in the US, and maybe elsewhere, it would be a positive item in your cv. I'll note that many, perhaps most, people in various fields who are able to go direct from graduate school to tenure-track positions have very little actual teaching experience. Whether such a direct move is common in your place and field depends on many things. I know that a typical fresh doctorate holder in Europe is likely to have quite a bit of experience already.
This varies by field, of course, and is possibly changing over time. As with any other application, make your case. No single thing will necessarily be the "slam-dunk" nor the door-slammer. You need to build a complete picture that points to your success and your willingness to learn and adapt.
But the postdoc itself and the teaching experience are both positive factors. Different institutions will value them differently, of course.