Can refusing to attend mandatory Title IX training be grounds for suspending a tenured faculty member?
Put yourself in the shoes of a university administrator. The federal government has made a law requiring your university to do certain things if it wants to maintain its federal funding. Those things are to some extent subject to interpretation, but according to legal opinions of your university's lawyers (e.g., the general counsel or other high-ranking legal advisers) requiring your faculty to undergo Title IX training about race is a good way to comply with the requirements and minimize the risk to your institution. So, you institute these requirements as mandatory. Some months later you are informed that a tenured faculty member is refusing to take the training.
What do you do? Select the answer that applies:
A. Invite the faculty member over for a chat in your office, where over cigars and a glass of scotch you both joke about the silliness of the federal requirements. You congratulate him on his good sense in objecting to these requirements, ask him to keep quiet about not taking the training so as to not make trouble for your university, and jovially send him on his way with an invitation to meet soon for a round of golf.
B. Invite the faculty member over for a chat in your office, where you tell him that you are dissatisfied with his disregard for university policy, but that you understand that he is objecting to the training on principled, conscientious grounds, and confide that you even agree with him to some extent. Taking that into account, and taking into account the fact that he is tenured and that you value his research contributions, you tell him that you have decided not to take any disciplinary action. You caution him to keep quiet about not taking the training and not encourage others to follow in his footsteps so as to not make trouble for your university, and send him on his way.
C. Send the faculty member a sternly-worded memo, with a copy to his dean and department chair, where you inform him that following his refusal to follow the mandatory requirement to undergo Title IX training, you are initiating disciplinary proceedings against him. You caution him that disregard for university policies on Title IX will not be tolerated under any circumstances, and that such proceedings could lead to various sanctions up to and including his termination. You conclude the memo by saying that until the proceedings have been set in motion some weeks from now, he has one last chance to satisfy the training requirement.
Now, it may be that you are the kind of person who would select answer B, or even A. I'm not saying it is unreasonable or wrong to be such a person, but if that is the case, I maintain that that would be in strong correlation to the fact that you are not a high-ranking university administrator -- let alone a successful one -- and probably never will be (I say this as a completely neutral statement, which I consider to be neither an insult nor a compliment). In practice, answer C is the only answer that a REAL university administrator (and certainly a successful one, who has a good ability to weigh the consequences of each of the different options on his/her own career and institution) will realistically choose.
To summarize, the answer to your question is an emphatic Yes: refusing to take mandatory training not only can lead to suspension or termination, but that is in fact the most likely outcome of such an action, by an overwhelming margin.
The short answer is yes, employees can be fired for not attending Title IX training. One example from Ohio State is here (the band director was fired). I could not find any examples in which a tenured professor has been fired for this. Failure to comply with Title IX is serious for any university receiving federal funding (which is essentially all universities in the US), so it would be an extremely serious situation.
Also, the premise of the question is misleading. It is not true that all university students and employees are required to attend Title IX training. From the US Department of Education's document on this
What type of training on Title IX and sexual violence should a school provide to its employees?
Answer: A school needs to ensure that responsible employees with the authority to address sexual violence know how to respond appropriately to reports of sexual violence, that other responsible employees know that they are obligated to report sexual violence to appropriate school officials, and that all other employees understand how to respond to reports of sexual violence. A school should ensure that professional counselors, pastoral counselors, and non-professional counselors or advocates also understand the extent to which they may keep a report confidential. A school should provide training to all employees likely to witness or receive reports of sexual violence, including teachers, professors, school law enforcement unit employees, school administrators, school counselors, general counsels, athletic coaches, health personnel, and resident advisors. Training for employees should include practical information about how to prevent and identify sexual violence, including same-sex sexual violence; the behaviors that may lead to and result in sexual violence; the attitudes of bystanders that may allow conduct to continue; the potential for revictimization by responders and its effect on students; appropriate methods for responding to a student who may have experienced sexual violence, including the use of nonjudgmental language; the impact of trauma on victims; and, as applicable, the person(s) to whom such misconduct must be reported.
So, the training is to teach employees likely to be confronted with sexual misconduct or receive reports of sexual misconduct how they should deal with that, including to whom at the university they should report the incident.
In particular, Section J deals with what training universities should provide.
Of course, you are free to do what you like, but I don't think you should advise people to refuse this training. They are putting themselves at risk of being fired over sessions that are designed to give them information on how to deal with a potentially traumatic situation. It seems a bit like refusing CPR training if the university decided to make that mandatory: I suppose the session is a bit of a pain, but is designed to help you deal with a situation in which responding appropriately could be critical.
I want to begin by saying that I don't have any direct experience with the firing of tenured faculty, nor am I intimately familiar with Title IX issues. In particular, the requirement mentioned by the OP regarding Title IX training for all faculty are not in place at my (public, US, research) university.
But I do have experience with the way universities and university administrators work. For instance, I had a colleague who in his first year did not want to sign a standard intellectual property agreement (which I have no memory of signing upon my arrival two years before, but I assume that I must have done it). It seems like a somewhat analogous question to ask: would the university fire this (untenured) person? The answer that I would give to that is "Eventually perhaps, but there would be a lot of intermediate steps, and the professor would have to be durably intransigent in a way that tired out the higher administration and their lawyers." In the particular case at hand, this issue went on for several months, and my colleague was satisfied by being permitted to sign an earlier version of the intellectual property agreement that he found more favorable.
As another example, every once in a while I am required to take an online test to make sure I am sufficiently cognizant of certain current laws: not related to Title IX but rather to FERPA. How does this work? In many stages: there is some announced deadline, and as it approaches I get reminders from someone in my department. If I miss the deadline I will get more personalized, urgent reminders. If I were durably unresponsive then (I am told) I will get locked out of access to certain student records: e.g. I would not be able to look up any student grades online. The idea here is that if my lack of training is viewed as a liability, the university will take steps to remove me from the specific situation that is causing the liability rather than suspending me generally. Of course, being removed from that situation may cause annoyance or hardship for me, and it may even interfere with my job.
Coming back to the question asked: in my own opinion it is very unlikely that a tenured faculty member would be suspended or fired for not complying with a Title IX training course...at least not as a first or even fourth step. Rather the university would on the one hand take steps to assess the liability involved in that specific faculty member not having the training, and if that liability feels considerable to them then they will try to reduce that liability while keeping the faculty member and/or squeeze the faculty member in such a way as to make carrying out the training preferable. By the way, there are many ways that an administration which is unhappy with a tenured faculty member can make that faculty member's life less pleasant which completely circumvent the expectations surrounding tenure: e.g. they could mess with the teaching obligations of that faculty member, regulate or limit their access to certain groups of students, give them less of an annual raise or no raise at all, and so forth. If the administration really thinks it is dead right and you are dead wrong, I think that they can, while keeping you around, make life more miserable for you than you can for them. In the case of the OP, he says that he will just leave. Exactly: that's a much more likely outcome.
I think that in order to consider suspension or firing in the short term, the administration would have to find some specific, actual legal liability rather than just a potential one. If a university fires a tenured faculty member for their own interpretation of a federal law, the probability of a huge fuss including censure from faculty associations and unions and/or a big lawsuit seems rather high. I would expect them only to risk that in the face of some other similarly proximate scandal or lawsuit. Another answer contains a link to a university band leader (untenured) who was fired for refusing Title IX training until after he mishandled a sexual harrassment / assault case between students under his direction. In that case they pretty much expect legal action either way, and they're weighing one lawsuit against another. If the OP has not had any actual dustups or issues with students, I would be absolutely flabbergasted (note: this could happen, obviously!) if he were fired rather than squeezed or forced out in a more subtle way.