This is probably a very broad question. It might be difficult to find one answer across all disciplines. If anyone could share their personal experiences on how they have designed tests and the thought process behind it, I would really appreciate it.
In the absence of other factors, in my opinion, at least in mathematics, a free response is always the more valuable type of question. The educational value of getting shown, where precisely your argument went wrong is much higher than just being told if the final result was right or not. It also allows for a much finer grading, as I can both give partial points for an almost correct reasoning leading to a wrong answer as well as deduct points for incorrect reasoning leading to the correct answer.
That being said, a test question never happens in a vacuum and the problem usually is lack of time. This can occur in two forms, lack of working time to correct the test and lack of time during the test.
Personally, I believe the former is a bad excuse. Unless you have a 200+ people class without any TAs (in which case, clearly something else is wrong), it is always possible to correct a full-text exam in a reasonable amount of time.
This leaves the latter point. Any exam has a limited duration and it might not be possible to cover all topics in this time. My personal response would be to leave out some of the less important ones at random. As long as the topic selection is not announced to the students beforehand, they still have to study all of them and the results will be similar. But this is a point where a good argument can be made for turning part of the test into multiple-choice. In this case, the most important topics should still all be free-responses, but you could add a small multiple-choice section covering the rest. However as these are not the important part of the test and they are there to save time, I would keep them simple. (E.g. picking the right definition or "does A imply B?") After all, if marking the right answer is not substantially quicker than writing it out, then you are not saving time.
In any case I would always invert the original question. Don't take a concept and decide of it should be multiple choice, but instead decide how many multiple choice questions you have to make and then pick the questions that work best.
There is a problem with multiple choice questions in general: They are very hard to design well. The time you "save" in grading isn't free. You need to spend a lot of time and thought on question design and you really need to find a way to verify the questions so that you don't get negative correlations between the answers and the knowledge of students. The latter has a statistical test, whose name I've forgotten after all these years. Sorry. Maybe someone can supply it. Some auto graders will supply that item analysis for you, actually.
In a free response question, the grader can apply judgement based on subtleties. In a MCQ, that all has to be designed into the question and the provided answers. And it isn't always easy. In a free response question, a small mistake in the question itself can sometimes be worked around if the student says something "sensible". There is no such chance in a MCQ.
Ideally, there is one correct answer, usually one clearly incorrect answer. But what of the others? Ideally, again, they should be designed to capture common student misconceptions so that feedback can be given to the student on why they chose the wrong answer and what they need to do to correct their thinking. Just an "incorrect" doesn't give any insight, especially if the incorrect answer are just chosen without deep thought.
If you aren't willing to spend an hour or more designing each question and if you aren't able to validate the test, then MCQs are very dangerous. The national level standardized tests do that sort of prior analysis and post validation. They also likely have a few surprises with new questions.
The statistical test I'm thinking of measures an individuals answer for each question compared to their overall score over all the questions and aggregates it over all the test takers. If the people answering question 1, say, do worse overall on that question than their overall score on the exam, then the question itself or its answers are likely misleading. Getting this feedback as the test creator can be quite humbling. Something is wrong, either in the teaching or the testing.
If you need to give MCQs for reasons of scale or otherwise, then expect problems. Try to build in some flexibility in the usage so that it doesn't punish people who actually understand things. This is very hard if good students "waste" time in considering weird questions when they should be doing something more productive. Students aren't always very good about picking the low hanging fruit first and then going back for the harder ones. Online testing might make that more difficult, actually as going back over skipped answers might not be well supported.
I assume it is still true, but at one time education degrees in the US required a course in test design. Given that, there must be text books devoted to it.
A great use of multiple choice questions is in formative assessments: you can test and quickly see which of your students have common misunderstandings. These questions don't work so well to assign grades, as then the student isn't interested in what they misunderstood, but are great for use in quizzes during teaching or between session to allow explanation of the errors.
As an example from a course I taught on the programming language R:
After running the following code, what will R return (answer without running)?
R will produce an error - Error: object ‘b’ not found
Each of the answers has its own reason for being chosen: 1 shows a misunderstanding of using local scope, 2 shows a misunderstanding of reassignment, 3 shows a misunderstanding of scoping going to global, and 4 is correct.
This kind of multiple choice question is obviously more work, but is much superior to a free response. I would do these with a show of hands and ask students to argue their point - much more difficult to do without the red herring answers.
There's nothing wrong with a healthy mix!
Though your question is very broad and the answer changes depending on the age group of students and their abilities, there is nothing wrong with a good mix (suited specifically to both of the possibilities above,) though in my answer I've included some facts and provided the most conclusive answer that I could.
While an open-ended question allows for the teacher to see exactly where the student went wrong and how to correct it, and also prevents a bad habit of guessing that often happens on multiple-choice, multiple-choice often has it's advantages as well.
For one, it makes for a much less tedious and painful, frustrating test. Sometimes students with special learning needs or learning disabilities may not be open to such a long test in this way, and if you are working with younger elementary-aged students, they do not have the attention span and it is often noticed that the quality of work goes down the more questions that are being asked.
However, know the abilities of your students. For High School/College level, more open-ended responses may be the way to get the test that gives you valuable information as a teacher. If you are teaching elementary students with a low attention span, maybe use one or two open ended questions with the rest being multiple choice.
Again, it really depends on the overall ability of the students. The starting ratio (for upper elementary students) should be about 50/50 (ex. 5 open-ended, 5 multiple choices) for a test with ten questions, but this ratio can be adjusted for the individual needs of your students or depending on the age.
I like multiple choice for Q's where you'd otherwise have to explain the sort of answer you want too much or you think they might misunderstand:
"What operation has the highest precedence? +, ^, *, /". I'm looking for a binary operator, not () or unary minus. Limiting the answers is the easiest way to specify that. "What is the formal name for a folder? Directory, file-bucket, Organizer, ...". Students can spin out on a term like "formal" and "the formal term that would be in the manual" is long and not all that much better. "Why did Virginia object to the 1st constitution? A, B ... ". There were lots of reasons. Writing "the best reason" tends not to help much. Choosing from a list with one clearly best answer is safer.
Of course, it depends on what you want to ask and how much detail. We could just as easily have them show the order of operators in an equation; or ask "briefly list 3 reasons Virginia...".
True or False?
Since there's a lot of intersubjectivity here, and I agree that multiple choice is not optimal for assessment, I notice that one aspect of multiple choice has been left out, so I'll play Devil's advocate: true/false questions.
While it does depend on your content area, adding a few well-written T/F questions to make up a part of an evaluation can be useful. It can save students time of writing out full answers better devoted to FRQs while still showing that they retained certain facts. It is up to the author of the test (that's you) to decide what merits a free response in a given subject matter. For example, in teaching world language courses, it can be a simple way to evaluate if a student has retained key cultural, geographical, or historical topics. Keep the statements simple and to the point (even drawing attention to negative statements). Even then, it can be better to ask a short free response question to eliminate guessing or cheating. Take things like really big dates or facts that students either know or don't (i.e. What year was the Battle of Hastings?), which do not benefit from a multiple choice question. Asking it as T/F doesn't ward off guessing and statistically encourages it. One deterrent for this is to use a free response or short answer question to spot check a random T/F question; this also encourages differentiated learning and critical thinking. Unless you are trying to get a computer to grade the entirety of an exam (don't; people take the tests and people can grade them), there's no more work for you to read "1066" than to see true or false circled. It is a better example of the student's work and retention. I would go with the short answer question there.
Think way back to your earliest spelling tests and which was the better assessment: did you show what you were thinking and how you were learning when you wrote out the word your teacher said or when you saw four words with three spelled close but wrong?
Overall, free response is the best insight and evaluation of your students; it leaves more room for the individual and less for testing errors. Of course, some disciplines and depths can fit in multiple choice better than others, but reading FRQs is not as tedious as writing multiple choice and a gives a better demonstration of nuances. I suggest sprinkling in just a handful of T/F for differentiation and time management.