What sorts of restrictions do universities place on romantic or sexual relationships between faculty and graduate students, and what are the underlying issues that motivate these restrictions?
For example, suppose the student has previously taken a class from the professor but has no plans for further academic contact. What other factors could help determine whether a relationship would be allowed or considered appropriate?
Official Answer: It depends on the tenets of your institution.
Real Answer: No. Never with an active student at your institution.
Relationships may sometimes be allowed between former students and faculty by the letter, but I cannot think of when it would be advisable for a faculty member to have a relationship with a student. Why not?
If there is ever a chance of you having power over their academic progress, it is at best a conflict of interest and at worst a setup for sexual harassment case. If you're looking for a great reason to have your tenure file silently quashed by a dean, this is a great way to do it.
Even if you have no direct power over a student, there is the possibility that other students could complain if there is indirect influence (e.g., if you have connections to other faculty who do have power over that student or over rival students). Even an indirect connection to the student has the potential for allegations of bias or preference.
You do not know the future and may run into a conflict later (e.g., student takes a course in your department, invited to work on a funded grant with member of their department). At that point, you're hosed. Even if ended the relationship as soon as the conflict became evident, the prior history of the relationship still exists. Alternatively, you could hurt your own (or the student's) academic trajectory by declining these opportunities. At the point where you are hurting a students' academic trajectory, it's unethical.
It looks bad on the institution. How many administrators would want it known that any of their professors are dating any of their students? If you were a parent looking at a school for your kid, all else being equal, would you want the one where faculty are dating students? Imagine if the relationship goes sour and you end up in shouting matches with a student in your office/lab (or theirs!). In that light, dating a student is an anti-service to the institution.
If the student is really going to be the love of your life, you can wait until they're graduated to start a relationship. If not, then it's certainly not worth the risk (even if your name is Professor Carlos Danger).
This sort of thing used to be allowed decades ago, where it wasn't uncommon for the (almost entirely male) faculty to end up marrying female grad students. As we've become more aware of the power structures and negative externalities involved, it's become much less permissible. Rightly so, in my opinion. No school is bigger than 100,000 students and most students you meet will be done in 3-4 years maximum. Given the risks (bias, bad press, potential lawsuits by the student or their peers), I don't think it's an undue restriction to not date a few thousand people rotating over a few years.