This post's basic claims:
If you 'should' feel bad about bad things (e.g. suffering), you 'should' feel bad about them in proportion to their magnitude.
Doing 1 is not a good idea, because a. it's intrinsically bad to feel bad, and b. feeling bad does not help you to fight bad things.
Therefore you should try not to feel bad about bad things, only to alleviate them.
All of which boils down to the proposition "It is instrumentally irrational to feel bad about bad things."
The problem is that 2.b. is blatantly false. Human beings are not capable of completely disconnecting action from emotion. Certainly, if you feel bad enough then due to the way the human brain works it's possible that you will (instrumentally irrationally) lapse into depression, and therefore do less to achieve your goals than if you never felt bad at all. And there are obviously cases in which what you feel emotionally can be overridden by calculation (e.g. trolley problem for many utilitarians). But given that no one performs explicit calculations to determine even most of their actions in more than a few small parts of their life, emotions determine most of our decisions. Do you really believe that someone who felt happy despite knowing about the state of suffering in the world would be more strongly motivated to reduce suffering than someone who felt a great sadness and a burning desire to stop it every time they thought about it? Do you think being happily mortal is the best emotional state for someone crusading to stop death?
If your actual goal is to end suffering -- if your moral system dictates, as most of ours do, that reducing suffering is currently by far the best thing you could do, and you actually want, unlike most of us, to follow your morals to their conclusions -- then you will do your absolute best to make your emotions about suffering dwarf all other emotions, because that is what will make you spend a life reducing suffering, and not any amount of abstract calculation.
On the other hand, if your real goal is to be as happy as you can, this post is great advice. But so is wireheading.
Do you really believe that someone who felt happy despite knowing about the state of suffering in the world would be more strongly motivated to reduce suffering than someone who felt a great sadness and a burning desire to stop it every time they thought about it?
Sadness and burning desire are not in the same bucket. Sadness doesn't make me want to do anything at all. It quickly leads to depression, which is about the most unmotivated it's possible to feel. Certainly when I'm happy, I'm more motivated to do something about saddening things than when I'm actually sad.
Now, anger is an extremely motivating emotion that might work better for these purposes...
Related to: I'm Scared; Purchase utilons and fuzzies separately
Expanded from this comment.
You have awakened as a rationalist, discarded your false beliefs, and updated on new evidence. You understand the dangers of UFAI, you do not look away from death or justify it. You realize your own weakness, and the Vast space of possible failures.
And understanding all this, you feel bad about it. Very bad, in fact. You are afraid of the dangers of the future, and you are horrified by the huge amounts of suffering. You have shut up and calculated, and the calculation output that you should feel 3^^^3 times as bad as over a stubbed toe. And a stubbed toe can be pretty bad.
But this reaction of yours is not rational. You should consider the options of choosing not to feel bad about bad things happening, and choosing to feel good no matter what.
Your bad feelings, whether of fear, empathetic suffering, or something else, are probably counterproductive. Not only do you feel bad - a loss of utility in itself - but such feelings probably hurt, rather than help, your efforts to change the world for the better.
You may believe that your emotional outlook must be "rational": that it must correspond to your conscious estimates of the present or the future. Perhaps you expect to die of old age, or perhaps you are aware of people being tortured in secret prisons. You are forcing your emotions to match the future you foresee, and so you feel unhappy and afraid.
I suggest that you allow your emotions to become disconnected from your conscious long-term predictions. Stop trying to force yourself to be unhappy because you predict bad things. Say to yourself: I choose to be happy and unafraid no matter what I predict!
Emotions are not a a tool like rational thought, which you have to use in a way that corresponds to the real world. You can use them in any way you like. It's rational to feel happy about a bleak future, because feeling happy is a good thing and there is no point in feeling unhappy!
Being happy or not, afraid or not, does not have to be determined by your conscious outlook. The only things that force your mind to be unhappy are things like pain, hunger, loneliness, and the immediate expectation of these. If you accept that your goal is to be happy and unafraid as a fact independent of the future you foresee, you can find various techniques to achieve this.
Unfortunately such techniques vary for different people. This post doesn't discuss any: it is about the prerequisite decision to be happy.
Expecting to die of cancer in fifty years does not, in itself, cause negative emotions like fear. Imagining the death in your mind, and dwelling on it, does cause fear. In the first place, avoid thinking about any future problem that you are not doing anything about.
Use your natural defensive mechanisms, such as of not acknowledging unsolved problems, or compartmentalizing different beliefs. Don't dismiss them as biases or irrational practices. They exist for a good reason and have their proper use.
This does not mean that you should ignore problems on the conscious level. It is possible to decouple the two things, with practice. You can take long-term strategic actions (donate to SIAI, research immortality) without acutely fearing the result of failure by not imagining that result.
When you're faced with something terrible and you're not doing anything about it anyway, just look away. Defeat the implicit LW conditioning that tells you looking away from the suffering of others is wrong. It's wrong only if it affects your actions, not your emotions.