In my country, in order to teach mathematics (or any subject) to college students one doesn't have to have a PhD. Just a good academic record in the masters level of one's subject and one must clear a (reasonably difficult) national level teachers eligibility test in one's subject. Having a PhD and research publications is a bonus but not mandatory.
So my question then is - would it be necessary to do research once you have secured a permanent faculty position? By necessary I mean, will I face any difficulties in teaching or explaining concepts and ideas to students if I don't spend some time doing research, publishing papers, learning and asking new questions in a given field?
One of the primary differences between tertiary education and K-12 is producing knowledge. At the K-12 level, teachers share knowledge but normally do not create new knowledge. At the university level, professors do share knowledge but they also contribute to the conversation by producing original research. In other words, the growth of knowledge is one of the main purposes of higher education.
Different institutions have different expectations of research at the tertiary level. However, it is often considered unusual not to take part in some form of research occasionally at the university level, even if your primary occupation is teaching.