Why do professors want to make sure that their notes written on the blackboard will not be published?
The scribblings a professor writes on the board during a lecture represent the lowest-quality type of content that that professor will ever present in public: handwritten, hastily created under time pressure and at a time of high cognitive load, messy, unchecked, and prone to error. On the other hand, professors, like any other professionals, will much prefer to be judged by the highest-quality content that they produce, e.g., their research papers, which are carefully crafted over a long period of focus and concentration, digitally typeset, polished repeatedly, and double- and triple-checked for accuracy. So, to the extent that a professor gets to control which of the content they produced gets publicly released, they will almost never voluntarily consent to the release of the low-quality content.

An analogy that occurs to me is that of a movie actor: for precisely the same reason as I described above, actors prefer for the public to form its opinion of them through their movies, and not through a paparazzi photo of them taken while they were out shopping for groceries, wearing disheveled clothes, no make-up, a sloppy hairdo etc. So, if you don't want to be like a paparazzi, you will respect the wishes of your professor and not publicly post photos of their blackboard writings.



Edit: I feel a need to clarify my remark about "lowest-quality type of content" and "hastily created [...] messy, unchecked, and prone to error". It seems that some people are reading this as a description of some kind of unprofessional professor who comes to class unprepared, delivers a poor quality lecture, then out of insecurity and fear of having their poor lecturing skills exposed publicly, refuses to allow pictures of their blackboard posted online. That is not quite what I meant (or at least is only one possible scenario covered by my answer). I was talking much more generally about any professor, whether excellent or poor, who comes to class, well-prepared or not, and delivers a lecture, which may be an excellent one, and then, for completely legitimate and rational reasons and not out of any insecurity, guilt, or shame, objects to having pictures of their blackboard posted online. 

Why would they object if their lecture is so good? Some have asked. The logic behind this is that the blackboard scribblings, even of an excellent and well-prepared professor, will still be a type of content that was created during a very short amount of time (the time of the lecture), while the professor is busy doing several other things at the same time (talking to the class, figuring out the details of the math or whatever it is they are writing on the board, consulting their notes, keeping track of time, etc.), is under the psychological pressure of being watched by a large group of people (many people find this a stressful situation), and has no time or cognitive resources available to detect or correct small errors that may be introduced inadvertently, no matter how carefully prepared one is. So, in a purely relative comparison between this content and other kinds of content that the same professor creates (e.g., research papers, which as I've said are thoughtfully prepared over many months), blackboard scribblings are relatively speaking a low quality content. It is simply an unfair competition: two different kinds of content, two different quality standards, and for the reasons I explained, many professors quite reasonably prefer to have only one of the two types be posted publicly online. 

Finally, note that all of this is not at all at odds with the fact that the blackboard writing can still be good or even excellent in the context of the lecture in which it is performed.