“I’ll go out soon” just does not sound like something a native English speaker is likely to say. It’s not idiomatic.
If you use leave instead of go out, it works better, but you would still be much more likely to hear it without soon at the end in the simple future (though it sounds very natural in the progressive future).
The difference in contextual meaning of these two types of future construction are far too nuanced and complex to boil down to just whether or not there is any kind of offence or resentment present in the speaker; so the answer to your question as a universal defining feature is no.
However, it is true that if you want to express resentment or irritation, it is quite likely that you will use the simple future, rather than the progressive future, at least within the type of context and verb that you’ve given here.
Thus, for example, the first hypothetical context for “I’ll leave” (a more idiomatic version of “I’ll go out soon”) that immediately came to my mind when I read your title was indeed one that can easily include some amount of irritation on the part of the speaker.
Imagine that you’re at work, and a colleague comes in and asks whether she can have her meeting with some client in the room where you are currently doing some work. Using the two versions of the same answer based on your question, you could then answer:
Sure; I’ll leave.
– or:
Sure; I’ll be leaving soon.
In the first case, you are saying that you will change your original plans in order to accommodate her and allow her to have her meeting in there: you had originally planned to stay in there working, but since she wants to have her meeting there, you’ll move somewhere else instead. This can be said quite neutrally, but it is very easy to instil some resentment into it as well (“Fine! If it’s so important to you, then I guess I’ll just go do my important work somewhere else!”) without changing anything much. A slight pause, a hint of curtness—that would do it.
In the second case, on the other hand, the implication is that you’re almost done with your working, so you were leaving anyway in a moment, and then the room will be free for your colleague to have her meeting in. Of course, it’s always possible to instil a tone of resentment into anything you say, but it’s much less likely with this construction. You’d have to do more than just add a slight pause for it to work.
So to some extent there is truth in the logic behind the explanation you’ve seen; but you cannot take it as a general guide to the difference between the two types of future constructions.
