I am looking for a word that describes the act of saying something like, "You shouldn't feel guilty," when what you really want (consciously or subconsciously) is the opposite. In other words, saying "Don't feel guilty" is meant to bring up the notion of guilt. I know there's a word for this but I can't remember it.
Try apophasis - Apophasis is a rhetorical device wherein the speaker or writer brings up a subject by either denying it, or denying that it should be brought up.
I don't have a single-word answer for you. Well, ok... I do, but it's pretty darn obscure.
So the answer that comes to my mind may not be what you're looking for. Generally one way of pointing out the insincere nature of instances like this is the pre-pend the word backhanded.
"Backhanded compliments" is the most common usage of this. (Perhaps the initiate phrase of such usage? Comment if you have research to add.)
But, "backhanded apology" and "backhanded comment" appear in general usage as well.
Some usages are more obvious than others, those tend to be the intentional ones (note the "Especially: Sarcastic" in MW). But the subtler insincerity of subconscious "passive aggression" speaks just as backhandedly as sarcasm ever could.
I also thought of backhanded, as well as insincere, pathological and my personal favorite, Machiavellian.
It is the substitution of a mild, indirect, or vague expression for one thought to be offensive, harsh, or blunt.
Other alternatives: Understatement, litote or antiphrasis.