How should a prof assign duty for a course with a discussion forum/board?
You ask &quot;How should a professor assign duty ...&quot; which is really an unanswerable question.  I think the real question is &quot;How can the team of TAs handle this workload in a way that is fair and not overwhelming?&quot;
As other people have said in the comments, I think you need to set some boundaries about amount of time, but also that you will not answer all individual questions. For example say there are multiple questions about problem 8. You do one answer for all of them. That's one reason not to rush in immediately, rather wait until you see what the pattern of questions are.
It's a little hard to address this without knowing the subject matter or the kinds of questions. Also in covid times it is important to be kind to and understanding of  students and the way that technology challenges are now a hundred times greater.  Further, problems that used to be solved in 2 minutes during a live class now take hours, both for you and them.
I know you are limited by what your supervisor will allow but I'd like to make some suggestions.   But before that I will also say that it is okay for you to set boundaries on when you will answer questions and response times. Don't be mean, but explain. For example, maybe a TA will come in at noon every day and try to answer questions, and another will come at 6.

First, discussion boards are not really a great way to do questions
and answers, and definitely not for expert answers (and the TAs are
the experts).  They can be great for peer to peer support and
that's what I'd suggest you encourage.  Can you ask your supervisor
if you could give extra credit (or even credit) for students who
answer questions?  Or even without that, can you encourage students
to jump in and help each other?  (I've found that my students can
often answer the technology questions really well, maybe they have
the same brand of phone or computer with the same OS which I don't
have.)

Second, notice on how Stack Exchange -- which is a set of sites
devoted to answering questions -- there are well defined rules
including obsessive refusal to answer duplicate questions.  Get
yourself and the students in the habit of searching for an existing
answer first. Teach them how to search effectively.
Also, write up a document &quot;how to ask a good question&quot; and revise it
as you get better at managing this. Put links to it everywhere.

Third, build up your documentation and knowledge base. You could
potentially use other features of your LMS for this (e.g. Blackboard
has so many ways of doing things that no one can use them all).  For
example, if the class is not using the wiki feature, start using
that. Or you could use Google docs. Or a lot of other things. You
could do FAQ but you can also start other things. Again, you can also
get the students contributing if it makes sense. Also if you are
answering questions that are strictly informational, post the
information prewritten (or less good, with well described links) at
the top of every page or in your signature. Try to do this in a way
that it can be used again next semester.

Fourth, teach students how to answer their own questions by showing
how to find information online or in their textbook or syllabus.
Many actually don't know how to use the index or table of contents of
the text books. Or, for example, &quot;I searched with keywords x, y, z
and found this useful video.&quot;   This gets to asking good questions,
but also, don't just answer, do things like say read page 329 and
then come back with questions based on that.

Fifth, once you have figured out how to actively sort questions where
students are really having complicated problems from the rest,
consider going into a meeting 1:1 or with a small group.  It's much,
much easier to answer the questions that way because you go back and
forth in real time.  If you record (and everyone involved is
comfortable) you can also post the recording of the answer.

Sixth, provide structure (and you may have to experiment with what
works.  For example set up separate boards for questions about
technology, questions about due dates and other syllabus related
issues, and then questions about each topic or chapter and then you
can have others like &quot;careers&quot; and &quot;random.&quot;

Seventh, have some prewritten templates so you can just fill in specific information.


I'm sure that there are lots of other things you can do too.