Does "je suis tué" mean "I am being killed", or "I am killed", that is, that I was killed and am dead or just about dead?
I would expect the answer to this question to apply to other phrases with similar constructions.
Out of context, je suis tué is rare because it needs to be pronounced while being dead1. It means "I am killed". It can be of course be used figuratively.
"I am being killed" better translates to on me tue but can also use the present tense like in je suis tué à petit feu par cette maladie.
Here are other sentences where je suis tué can be used:
Je tue ou je suis tué - I kill or I am killed.
Si je suis tué au combat,... If I'm killed in action...
1Paradoxically, je suis mort is not affected by this issue and is a common sentence.
Je suis tué is a passive voice. If you had used any other subject pronoun than the first person I would have little hesitation on explaining the meaning of the sentence. If someone is killed (tué), they're deprived of life, and unless speaking from an afterlife they can't utter those words.
You are not giving the context in which you read/heard the sentence. As @jlliagre says it could be used with si to express a condition in which the speaker imagines they are deprived of life.
It is also likely the verb is used in a figurative way1 to mean "shattered" or "strongly impressed". Like in:
It also tends to be used in place of mort in phrases like:
Here are two examples where both meanings overlap:
In this Je suis Charlie poem:
Je suis tué nous sommes tous aujourd’hui Charlie !
Where it takes the meaning of the actual killing in the massacre, and the moral killing of those speaking and mourning the deaths.
Je suis tué par votre silence.
Where the homeless who's speaking is wrecked and will eventually be destroyed because of people's indifference to his plight.
1 Just like kill in English