Is it okay to post textbook solutions online? It seems to me, based on two previous stack exchange questions (1, 2) that this is legal, but I'm wondering more if it's "okay," i.e. what do academics/professors think of this?

I'm interested in posting (my own) solutions to textbook problems (that I find interesting) online, as I self-study them. I'd guess some problems may be of interest to others, I get to possibly have discussions about them (maybe I mess up), and I get to practice explaining technical concepts through text (I'm interested in pedagogy). For what it's worth, I'm an undergrad (though soon to be grad) student. Ideally I'd want to do something similar to physicspages.com, though I doubt I'd have the same rigour. I'd be working on a textbook myself and using this as a sort of "learning by teaching" approach. 

On the other hand, I know many professors use textbook problems as homework, and a publicly displayed website with textbook solutions might be easy to find on a search engine (so students may just copy). Would instructors think this is inappropriate? 

In case it matters, I intend to start with Topological Insulators and Topological Superconductors by Bernevig since this is likely to be related to my PhD topic. (Though this could change on a whim)
While I don't think there is anything formally standing in your way of doing this, as long as you don't post the questions themselves or answers from the publisher's answer key, I don't think that it is a wise thing to do. 

But I'll focus on future students rather than professors to explain why it isn't wise. For many things the only real way to learn them is to practice. This is true in math and the sciences as well as in, say swimming or chess. Learning involves changing the brain. See, for example, the book The Art of Changing the Brain by James E Zull.

The implication of this is that, very often, the most important part of a published textbook is the exercises, not the explanatory text. Students need to, perhaps, struggle with those exercises to learn the material. If you give them a shortcut to getting the answer, while it feels good to them, it makes it harder for them to learn the material. Too many students don't understand this (hence the need for a book like Zull's). It isn't a question of being able to turn in "homework" as much as it is having useful practice exercises that you can't short-circuit. 

If you want to meet your own goals and also publish something useful, publish a set of additional exercises for a good book instead of answers for the existing exercises. You will find (a) that this is harder and (b) that it increases your own knowledge by changing your own brain. 

You could also, as some publishers do, publish solutions to half of your questions. Some textbooks, for example, have solutions to, say, the odd numbered exercises in the backmater. 

You might even get an offer from a publisher to do that for other books. Some textbooks, in fact, have the questions provided by other than the author. 

But publishing answers is a disservice to other students, not a service.