The early works
Kant gives multiple explanations for why lying is wrong and these accounts differ. In Kant's earlier works, Kant gives several different arguments against lying. Arguably the first in the Groundwork occurs before he even articulates the "Categorical Imperative" when discussing actions, he considers why a shopkeeper should be honest. Kant dismisses the idea that he should be honest just because it's good business.
Moving forward, Kant gives two reasons that parallel articulations of the categorical imperative:
Version 1: Contradiction in thought
In his Groundwork, Kant's first argument is that when I think of lying as being a universal necessity (thinking my maxim to lie as a universal law according to the Categorical Imperative), the thought contradicts itself as I am dependent on others to believe my lie in order to have my maxim work but when I test it through the Categorical Imperative, I become aware that when it was a general law to lie if it fits myself, no one would believe my words:
I soon become aware that I could indeed Will the lie, but by no means a
universal laW to lie; for according to such a laW there Would actually be
no promise at all, since it Would be futile to pretend my Will to others With
regard to my future actions, Who Would not believe this pretence;... (Ak. 4:403)
The same argument is reiterated in regard to false promising, which is equated with lying before (Ak. 4:419 and 4:424):
when it is said that you ought not to make deceitful promises; and one assumes that the necessity of
this omission is not merely giving counsel for avoiding some other ill, so
that what is said would be: you ought not to make lying promises lest
the universality of a law that everyone, once he believes him
self to be in need, could promise Whatever he fancies With the intention
not to keep it, would make the promise and the end one may pursue
With it itself impossible, as no one would believe he was being promised
anything, but would laugh about any such utterance, as a vain pretence.
This is what he calls afterward "contradiction in thought" (Ak. 4: 424). Although this is a nice argument, it is totally unclear how to handle cases of a lie by omission here. Therefore, we have to go further.
Version 2: Violation of the humanity of others
Another argument, which I highlight because it is parallel but different from his later arguments, is the claim that it is wrong because it does not respect rationality in others. And this specifically relates to telling lies rather than omitting information. On his most basic picture, people are to be conceived of as rational and good even if empirical evidence says otherwise. (Conversely, we know that we ourselves are evil in his view). Here, Kant argues with the formula of humanity (Ak. 4:429):
So act that you use humanity, in
your own person as well as in the person of any other, always at the same time
as an end, never merely as a means.
When discussing again the example of false promising, he argues that it would violate the ultimate value they represent (dignity) by virtue of sharing the faculty of reason, i.e. rationality. It would use a rational being merely as means to an end, i.e. instrumentalise them (Ak. 4:429):
someone Who has it in mind to make a lying promise to others will
see at once that he wants to make use of another human being merely
as a means, Who does not at the same time contain in himself the end.
An omission also seems capable of leading someone to miscalculate, but the trick is that one would not necessarily be merely the means of the person omitting informations. Instead, they could be acting carelessly on their own.
A practical example might help here. Imagine evil-me and nice-you are standing outside of a building. We work as demolitions experts.
Scenario A
nice-you: did all of the children leave the old orphanage so we can blow it up?
evil-me: yes.
nice-you then blows up the orphanage.
This is my fault, because you acted on my information that it was safe.
Scenario B
nice-you: did all of the children leave the old orphanage so we can blow it up?
evil-me: I know all of them are out of the west wing [omitting that I know three kids are in the east wing].
nice-you then blows up the orphanage.
This is on you, because my information did not indicate it was safe.
The Metaphysics of Morals
Kant in the Metaphysics of Morals: Doctrine of Virtue (not to be confused with the Groundwork) interestingly suggests a different reason why lying is wrong. Here, he suggests a violation of one's own rational nature. Because to tell a lie is to cease to behave rationally [1]. He also talks of "truthfulness" rather than "telling the truth", which in German ("wahrhaftigkeit") is destinct and is more in line with the latin quote included (Ak. 6:429):
The greatest violation of man's duty to himself regarded merely as a
moral being (the humanity in his own person) is the contrary of truth
fulness, lying (aliud lingua promptum, aliud pectore inclusum gerere*). [...] By a lie a man throws away and, as it were, annihilates his dignity as a man.
*"To have one thing shut up in the heart and another ready on the tongue." Sallust,
The War with Catiline X , 5 .
[1] An important point here is that what Kant means by "rational" doesn't just mean "behave reasonably" but is linked a deeper idea of being a specific type of being that has a rational nature that is not defined by the phenomenological world
Now, the case of carefully planned omissions isn't going to hinge on casuistical questions Kant considers in MPV. Kant seems to accept some. For instance, he accepts that in polite society, we don't have an obligation to tell people things like "your face is ugly," but he doesn't address every type of circumstance.
Conclusion
It's clear we can on his view omit information in part because the duty is not to truth-telling simpliciter but to never mucking up my rationality or the rationality of others by providing misinformation.
At least since 1787, Kant distinguishes not between telling the truth and lying, but between truthfulness and lying (see 5:61). This is more in line with the argumentation he later carries on using (humanity) and is to some extend even explicit towards malicious omitting as being a form of lying in contrast to omitting things e.g. out of kindness.
(We could also add further references from Kant's Lecture on Ethics that show how his view contrasted with earlier rationalist morality).
