Como dizer "Eu ando odiando o meu novo emprego" em inglês

Brother,

Os exemplos que vocês nos trouxe na pergunta estão corretos!
Verb BE + verb HATE + ING

Indeed "hate" is a stative verb, so we don´t usually make it a gerund in sentences with continuous tenses .
So, no "I am knowing/loving/thinking/understanding" etc, in such context.

You could express that with something like "my job is making me miserable/I can ´t stand my job anymore/my job is making my life miserable/I hate my job (then elaborate more if you wish, but that could give you a headache should your employee lurk around, or some brown nose...)/my job makes me feel emotionally bad/I don´t like my job (anymore).
Plus - "I am getting more and more frustrated about my job/...frustrated with this job/my job sucks, I am getting sick of it (ficando farto/farta...)"

And other ways that could express that.

Ah, where it reads "my job" should read "my new job".

There's nothing wrong with saying "I've been hating",actually. Off the top of my head right now, I can only think of one verb that doesn't sound right in this context when it's in the gerund form which is "know". But again, in this context in which the verb is being used, of course "know" can have the "ing" added at the end of it in some other contexts and sentences. In portuguese one might say "estou sabendo" and "estou conhecendo", but that doesn't work in English, to express "estou sabendo" in the sense that they've been made aware of news of some kind they say "I've heard" and "estou conhecendo" as in they"re getting more familiar with something or someone, they say " I'm getting to know (it/her/him/them) better. But it doesn't really apply to "hate", feel free to put it into the gerund form when the context requires it. And also keep in mind that i'm not a grammarian of any kind so this is just based off "because it sounds or wrong", just like an average native speaker with no deeper knowledge of grammar would do their teaching in an attempt to explain things like this to somebody, so don't take my word for it if you want to hear it from someone who knows the actual rule that backs this up.

Oh and "agree" also falls into the same category as "know". But then again, at no point did I say that "know" is the only verb that can't be used like that, some other ones start popping up as you look deeper into your head, it's just that laziness gets the best of you sometimes, doesn't it?

Chill out Leo, don´t sweat the small stuff.

Actually I was thinking in terms of "I am (loving/knowing/hating), that would be unnatural, not advisable. Now I read that it is "I have been...", so it´s acceptable in some given contexts - like the one of the initial sentence.
The verbs I mentioned were all for the sake of illustration, so not having to do with their use here.
Thanks for your comments, insights and participation, they are always welcome.

A fim de agradar gregos e troianos, uma opção é utilizar:

My new job has been getting on my nerves. [Eu ando odiando o meu novo emprego.]

Bons estudos.