I wasn't expecting so many answers to this question and it took me a while to process each of them. I chose to accept the answer I did because I consider it to have most closely resolved the entire issue. My answer here goes through and aggregates the relevant information from across the answers so as to directly address the questions and examples in my original question and help explain why I chose the answer I did.
The first notable problem was question's misidentification of truth as the relevant term. This is admittedly my fault. As it turns out, the truth is more relevant in the first and second examples:
I care about the truth
He speaks the truth!
While antonyms to truth include words such as lie or lies, they don't work in the original context:
I care about the lie
I care about the lies
Each of these mean something drastically different from my original intent. Namely, the concept of the truth is different from that of a truth. Lie parallels with a truth. This does, however, make it applicable for the second and third examples which could be why so many answers focused on this style of antonym:
He speaks lies!
This one lie is important to all
But it again doesn't quite convey what I was looking for. As such, I am unsatisfied with lie as an answer.
Falsehood is much, much closer:
I care about falsehood
He speaks falsehood!
This one falsehood is important to all
Unfortunately, most suggestions of falsehood were paired with lie and it isn't obvious that falsehood also works as an antonym to the truth. I wasn't able to determine if people were thinking of these examples instead:
I care about falsehoods
He speaks falsehoods!
Which again brings us to the problem with using lie. While falsehood would work, I don't think people were realizing why — or, at least, they weren't explaining why.
The major alternative to falsehood was falsity:
I care about falsity
He speaks falsity!
This one falsity is important to all
This is dead on in meaning and is included as a noun under the definition of false. Falseness also happens to be there: "I care about falseness."
All of this heads into a clash between three acceptable terms: Falsehood, falsity and falseness. The last two seem a little outdated or unused and the prevalence of answers suggesting falsehood tilt acceptance toward an answer that includes it.
Unfortunately, the only answers that explicitly note falsehood being usable as "I care about falsehood" were a short answer that only said this. The highest voted answer merely had the word, a link to the dictionary and included lie which I consider an incorrect answer (through no fault of the answerer; it appears that my question was unclear in its goal.)
In my opinion, the answer chosen deserves it because:
- it mentions both falsehood and falsity
- it discusses both "truth as fact" and "truth as reality"
- gives a nod to lies and their synonyms
Of note, one editor pointed out that "the false" is acceptable with regards to the philosophical concept. This again matches perfectly with the desired meaning. I did not accept that answer, however, due to its limited scope and other answers contesting that "the false" is incorrect or ungrammatical.
And, in other news, it turns out the word bullshit works surprisingly well.