When I heard something really bad and I felt hurt, so I say, "I couldn't not feel hurt after hearing that." meaning "I couldn't hear that without feeling hurt." Does the sentence in bold sound natural at all?
People will understand this as a humorous way to express the idea: I couldn't not take an offer like that = I just had to take that offer. I'd choose a more straightforward way to express something I felt strongly about: "I couldn't stand to hear it" or "It hurt me to hear it".
This is an excellent way of expressing yourself, especially in response to someone who thinks you are over-reacting to what you heard. It is equivalent to saying, 'any reasonable person would feel hurt by that'.
This double negative - 'couldn't not' - is primarily used when something really good is presented to the person, and it is too good to turn down, too good to just let go.
My car dealer sent me a letter. Mazda would pay me the going trade-in value for my car, and deduct another £5,000 off the price for a new one. So, I could get a brand new car for £6,000. It was an offer too good to refuse/ I couldn't not snap it up.