Dead Week has a set of formal rules for professors, but they are rarely enforced and often not well-understood (by faculty or students). Faculty and grad instructors seeking clarification generally get contradictory and confusing information.
In general, after teaching several classes independently, the rules are (and they do vary some by discipline - notably, labs and some other STEM/applied sciences courses necessarily require somewhat different guidelines.):
No tests, no quizzes allowed during Dead Week (unless they are 'normally assigned' - e.g. it is okay, for example, to do a reading quiz if you give a reading quiz every week)
Absolutely no final exams during Dead Week
Homework assignments are only acceptable if they are normal, similar to quizzes. Moreover, they should be short (and generally should just be avoided). However, having one due during dead week is fine.
In the past decade there have been few, if any, actual attempts to enforce the rules by the university... and this may be, in part, because professors may technically have a right to ignore the rules - professors are determinative of individual final grades without exception, unless there is a grade appeal which is something a student may win, even likely to win, but which is a huge hassle for all involved and does more damage to a student's reputation than the grade would have in many cases. Professors often feel forced to make all final projects due by dead week or may feel it necessary to shorten deadlines on final assignments so that they are due prior to dead week. Dead week essentially makes one week of material less important as material covered during it can't be graded outside of a final exam - dead week classes may feel more boring as a result while the previous week felt overly rushed with too much material. Professors may be wary about giving the final of, say, 3 or 4 tests during dead week and instead have their final test during finals week instead - even though it isn't a final exam in the same way a comprehensive, extra-long and more highly-valued test during finals week tyically is. This can mean students have to stay on campus longer or actually stress more.
There are workarounds that professors and instructors can, and regularly do, use as well. Give the final in class and refer to it as homework, like other homeworks they've assigned, but 'worth double'. Essentially, playing semantic games can make it easier to give a final during dead week - or to give extra assignments.
Online finals are becoming more common, and not just for online classes - many in-person classes are now having all their tests online (far less time consuming to grade for the professor, students get grades back immediately, virtually no chance of human error in grading, more flexibility for students). Professors can just make them available during Dead Week and give students longer to complete them. Some departments however are wary of even making a final available during dead week - though this goes against the purpose of dead week which is to reduce stress and allow students time to study and prepare according to their own schedules.
This isn't dissimilar from a take-home final to be turned in during finals week (note that a take-home final cannot be both assigned and made due within the span of Dead Week - technically, it shouldn't be due during dead week at all if it is a true final, but if your final is a research paper you've been working on for a month, or a final project, then I wouldn't complain - remember, your professor also needs time to grade finals and if your final is a research paper, then they have 50+ research papers to grade during finals week).
In such cases you should know you have the right and are entitled to have up until the time when your exam for that class would have been held based on the exam time assigned to it. This is an important student right to be aware of, but it also faces new problems: