This is advice I have heard primarily when it comes to speaking to children, both in private and in the classroom. For example, this article on using positive phrasing. This advice is most often given in relation to advice/commands for children, rather than just descriptive conversation. In fact, the first time I heard this advice was in a Red Cross baby-sitting class I took as a middle-schooler (not recently).
The general justification is that A) the positive phrase gives a clearer explanation of what your expectations are and B) the "not" in your sentence will often be drowned out in children's hearing by the other words in the sentence. So in practice, you should say
"walk in the library"
vs
"don't run in the library"
because the first is unambiguous, whereas the second might be heard as either
A) "hop or skip or dance in the library, so long as you don't run"
or
B) "don't RUN IN THE LIBRARY!"
I suspect the same might be true when speaking (or writing) to adults. Consider Nixon's famous self-defense--I personally hear it as something like
"I AM not A CROOK"
even though he didn't actually say it that way. For good or ill, I can't imagine that it would have been so viscerally memorable if he'd said instead "I am an honest person" or some such.
On the other hand, there definitely are situations in which the negative construction is subtly better or more accurate. "Not guilty" vs "innocent" immediately comes to mind--in the US jurisprudence system, a defendant is only ever found "not guilty", never "innocent", because what the jury has actually decided is that the state has not presented enough evidence to convict on the particular charges laid, not that the defendant is definitively, factually innocent of the crime, and certainly not that the person is generally "innocent" in the Eden-before-the-fall sense.
In classic formal logic, the opposite of A is always "not A"--which means the universe of possibilities that aren't A--rather than Z or whatever we think of as the polar opposite. Thus the opposite of up is not up rather than down (leaving room for the possibility of at the current altitude and such); the opposite of black is not black rather than white (leaving room for more than 50 shades of gray plus red, green, polka-dotted...), etc. We usually don't think in those terms in regular life, but the subtle distinction can still be important. When giving orders, we don't want ambiguity; but in other situations, we might want to leave in the possibility of more than one alternative to whatever our "not" is or change the emphasis to the thing being negated rather than the potential positive alternative.
For example:
Q: "Where do you want to go to lunch?"
A: "Not the cafeteria."
We could replace the ambiguous "not" statement in the answer with a definitive positive statement about McDonald's or the Russian Tea Room or what-have-you, but then we've significantly changed the meaning of the exchange.
In your examples, none seems to be a clear-cut case where the second alternative is better. "Not true" can mean a variety of things in addition to false--partially true, an outright lie, a mistake, etc. Depending on context, we might want to use one of these explicitly, or we might want to leave it at "not true."
"Infeasible" seems to be an example of concepts that are only expressed as negatives--this is really just a Latin-ish way of saying "not feasible" rather than a truly positive statement. I can't think of a way of saying this that doesn't include a negation, like unworkable or impractical. The best Thesaurus.com comes up with is "absurd," "futile," or "preposterous," each of which I think is a step past your original sentence. Infeasible also has the drawback of the "unfeasible vs infeasible" debate, so "not feasible" might well be the more felicitous choice.
I agree with the comments that "not useful" is different than "useless," though perhaps here you could say "the printer is broken" or "the document must be handwritten" to better effect.
