Relevant meanings of "litany":

1 : a prayer consisting of a series of invocations and supplications by the leader with alternate responses by the congregation 

2a: a resonant or repetitive chant

The point of reciting this kind of litany is to calm yourself and remind yourself of the right way to approach things.  This particular litany seems like it would be used in situations where you've learned something horrible and are thinking, "Oh god, why did I look there/why did he tell me?  I regret learning this."  It would be unfortunate if emotional reactions like that led people to punish themselves or other people for passing on true knowledge.

I would say that, in the worst case, if you think for a bit and can't figure out anything useful to do with the new knowledge, then you can just continue acting the same way you were before, and thus be no worse off.^[1]  If you can gain control of yourself, that is.  If you can control yourself, then you won't be worse off, and then the litany is true.  In that sense, I think it's meant to be a self-fulfilling prophecy.

Compare:

I will face my fear.
I will permit it to pass over me and through me.
And when it has gone past, I will turn the inner eye to see its path.
Where the fear has gone there will be nothing. Only I will remain.

This portion of the "Litany Against Fear" is very obviously meant to be a self-fulfilling prophecy, where the process of reciting it helps make it true.

1. ^{^}

   Possible exception: if you have an honesty obligation to pass the information on to someone else, who you expect to then do something like break up a relationship.  There are a few ways to approach this.  One is to say, well, if you didn't know, then you'd be living a lie, and is that what you want?  You think you wouldn't find out eventually?  Another is to say, if the parties involved want to have honesty obligations, then might it also be reasonable to have "Try to reward me and not over-punish me when I tell you a difficult truth" as a principle?  (It might not be enough in all cases to successfully incentivize truth-telling, but I suspect a lot of people could do with some steps in that direction.)

As @localdeity said, you won't find what you're asking for, because it isn't strictly true among humans. The way our minds are constructed, we are not equipped by default to accept arbitrary information and interpret and apply it effectively. There are theorems about how, in the limit of infinite computing power, a mind that correctly updates beliefs in response to evidence should never become predictably worse off (either epistemologically or in its ability to make plans to achieve goals) as it acquires new information, but those theorems don't strictly apply to our own bounded rationality, heuristic-laden, limited minds.

That said: it's true enough that it's worth having a very high bar against trying to force yourself to believe in falsehoods you think are useful, or convincing others to do so. Even if you can do such a thing, it's dangerous in a variety of ways, some obvious and some not so obvious. It's definitely worth knowing that even when a fiction is useful in the short term, it's almost always the case that once you recognize it as such, you as an individual are going to be better off admitting it and getting through to the other side where you understand why it is a useful approximation or pragmatic implementation of some deeper and more-true principle.

I've always regarded the Litany as normative as much as descriptive. If you find yourself flinching away from discovering a truth, you are doing something wrong.

I think you misunderstand litany of gendlin. 

https://www.lesswrong.com/w/litany-of-gendlin

Let me go line by line, on a fairly accurate interpretation I buy into. 

What is true is already so

I think this is obvious, if you get past the linguistic confusions, what is true is already so. 

Owning up to it doesn't make it worse. 

This is basically asserting that map is not the territory, acknowledging what's true doesn't make it any worse—aside from few minor exceptions where map and territory intersect. (See placebo) 

Not being open about it doesn't make it go away. 

And because it's true, it is what is there to be interacted with.

Yet another reassertion of map is not the territory. A person deals with consequences of how reality is, regardless of whether they believe in it. True beliefs predict experience (can be interacted with). 

Anything untrue isn't there to be lived. 

Well if true roughly is {what predicts experience as a result can be interacted with} then ~true is {what doesn't predict predict experience as a result cannot be interacted with}  

People can stand what is true,
for they are already enduring it.

Well the thing is people are already dealing with consequences of what's true or living in a world that is in causal web of what it entails, changing a belief which allows them to predict those consequences better isn't going to change anything of value, they're already enduring all the worse experiences which that truth can bring them in their daily lives. Remember edgan's law , our theories about the world only add upto what was normal day to day experiences and intuitive models predict, nothing much changes they just get a better way to predict it. 

If let's say all my family members died yesterday, and I am a unemployed, I would be dealing with the consequences, regardless of whether I delude myself on whether they're alive, for one accepting they're dead will allow me to predict those consequences, besides I would —within that delusion— either become totally mentally deranged unable to endure the tragedy  or as the litany says, I am still enduring its consequences and end up with a belief in a belief about my parents, which is basically same as not being deluded just being mistaken about my beliefs. 

This line of litany brings clarity to the person who is trying to hide away from their own beliefs or are mistakenly blunting the predictive power of their beliefs in hope that it will change or help with the consequences they have to endure. 

So in summary, yes some experiences can be bad that people cannot endure it, but if they happen to be enduring certain experiences and intuitive models, they can stand better and more true models.