As a professor, can I copy exercises / problems from a textbook without referencing the source? In creating a final exam for my course, I would like to use problems from a textbook that I did not assign to the course. Do I need to reference the source on the final exam paper or is it not necessary?
I have never seen a university exam in which problems are cited to sources.  The vast majority of exams I've seen are intellectually derivative on other sources: for many courses, an exam which is not intellectually derivative would be difficult to write and very painful for students to take.

In my opinion, in current academic culture course exams are not viewed as academic works which are subject to the standards of plagiarism: there is no pretense of originality.  However, copying exams from standard sources has issues other than intellectual priority: as already mentioned, it may give an advantage to students who are consulting those sources.  In my opinion, it is a good practice when writing an exam to never copy a problem wholly and directly from a standard source.  A question can always be reworded.  For a large array of problems in STEM fields, parameters can be adjusted or other incidental features changed, creating a problem that to a sufficiently expert eye looks isomorphic to the textbook problem, but is different enough so that someone who would feel that way has mastered the material.  

When I write undergraduate exams, I usually do so with my own study materials and exams given in past iterations of the course in view (and I also give these to the students, in most cases).  With this in view, I try to create an exam which has the right amount of similarity to past exams and review materials: neither too much nor too little. For intermediate undergraduate level math courses, I have found that having one problem or part of a problem be basically identical (though somewhat reworded) to something in the study materials has a positive effect on the course: as the students run through several midterms and the final exam, they learn that the study materials are actually to be carefully studied and mastered, which increases their learning.  In general I find that changes that I regard as relatively modest -- e.g. swapping out one problem for a "cognate problem" -- are often regarded as "completely different" by students.

In short: you can and probably should use other sources for inspiration and calibration purposes when writing an exam, and I have never seen anyone document their sources as if an exam were an academic paper.  (In most cases one would not be documenting the "true author" of the work but simply identifying the previous person in the borrowing chain.)  However, you should not blindly copy problems from other sources, and you should watch carefully to ensure that your students regard your exams as having a sufficient amount of novelty.  To do otherwise is to shirk one of the most basic duties of the course instructor.