I submitted a paper to a journal where four reviewing cycles were done by two reviewers. One of the two reviewers had no comments and accepted the paper as is, while the other reviewer had comments at each revision cycle. All comments were addressed. Unfortunately, after about a year, the paper was rejected. Subsequently me and the co-authors decided to write a rebuttal letter to the editor in chief. It has been two months since our rebuttal letter but there has been no reply from the journal. Should we write a reminder, inquire about any updates, or just patiently wait?
From the point of view of the journal, the case is closed with their rejection. Of course it would be a decent thing to respond to your rebuttal, but they are under no obligation to do this, or anything more. They have done their work reading and assessing the submission and reviews and have come to a final decision that is meant to be really final. Opening the process again is highly nonstandard (it has happened occasionally but I'd give that a subjective probability of 10% or so, based on my own experience as editor and author). So there are no "updates" to expect and a response may come at any point in time or not. Note that many editors are not paid, may be inundated with submissions, and have many other things to do, so they may not be keen on dealing with a paper after the process is closed. (Personally I'd probably find the time to write at least a small response but it is not strictly part of our job to do that.)
You should definitely not make any plans to submit elsewhere dependent on receiving a response to this. In fact if your idea was to wait with any submission elsewhere until you have a negative response for this, this was probably not a very good idea, because you may not get an answer. (Of course the hope was to get a positive response, and that'd have been a possibility but unlikely from the start.)
Not long ago I was part of a team of authors and we had a paper rejected, and some co-authors suggested to write a rebuttal. I was against it for the reasons given above. The majority of authors voted against me, so we did it anyway. After four month we got a reply that confirmed the rejection and didn't really add anything informative, so sending a rebuttal and waiting for the answer basically cost us four months.
That said, at this point writing a reminder email is maybe good because it increases the probability that you get a faster response (which you probably feel you need for submitting elsewhere), even though probably negative. An alternative would be to send them an email stating that you have decided to submit elsewhere and you don't await a response to the rebuttal anymore. At some time point this will be your option to "cut your losses".
It's normal to respond to the first rebuttal. If that interaction turns out to be unproductive, then they might stop responding. The exception is if your rebuttal itself is unproductive, e.g. if it contains insults or personal attacks.
So yes, assuming your rebuttal was sent in good faith, two months should be sufficient for them to respond. If they have not, it's reasonable to send a reminder.