It depends on whether I trust a teacher or not or how I'm reacting to them, what questions I'd ask them. Sometimes I don't ask any questions of some teachers because due to my listening to and examination of what they are saying, I don't have confidence in them. Other times if I like what I hear, I may ask questions to examine their different perspectives of certain issues, but it would really depend on what they had been saying.
One question that is a favorite of both Steve's and mine, is to ask them how important they feel the quality of compassion is in the practice. This is quite an important question we ask teachers so that we can see whether they understand the importance of compassion is in the practice, especially in developing Right Intention, or whether they feel that we don't have to practice it, and that it will just come by itself later.
Another one of our favorite things to ask them how important they feel reflection-type meditation is in the practice. If they feel it has no importance whatsoever, that we just have to only be mindful and concentrated, we usually realize that we do not really have much in common with the teacher. However, when we meet a teacher who thinks reflection meditation is very important, we often ask them a lot of questions.
Sometimes the questions we ask them depends on what is happening in our own practice. When we trust a teacher, then we may ask questions about that. But it is important to trust a teacher before you ask them personal questions.
Once I asked two different teachers the same question, and the different responses were very interesting. The first teacher had taught us quite a bit about wise reflection, and I asked him about confidence-inspiring objects of meditation. He gave me quite a detailed answer. I was very impressed with his knowledge. I asked the same question to another teacher, and he chose not to answer it in that way and instead mounted a personal attack on my character, telling me that if I had a lack of confidence I would have a problem. It got so bad that Steve intervened to change the subject. So I realized that this teacher was not really for me.