What does Kosh mean and what is he talking about when he says that the Centauri and Narn are dead (or does he say dying?) races? Is it because they are so embroiled in war and hatred of each other they'll destroy each other? Or is there something else effecting them?
Looking at some season 3 quotes from Kosh here, I will try to address this.
Consider this scene in Dust to Dust.
Note that this is a vision G'Kar has, so this may not be Kosh. It is fitting that it is Kosh though. Kosh says "We are a dying people, G'Kar. So are the Centauri. [We see him, standing behind G'Kar: an older Narn, not G'Kar's father.] Obsessed with each other's death until death is all we can see and death is all we deserve."
In effect, yes the Centauri and Narn are locked into a cycle of destruction that they will not break if they continue on as is. E.g. obsessed with "honor", with blaming the other side, with being "in the right". Everything Kosh says to G'Kar could have just as easily been said to Mollari. G'Kar's response is proof of this; he blames the Centauri ("they started it"), he discusses honor and the need to right previous wrongs.
The parallel is obvious:
The Shadows and the Vorlons are locked into an identical struggle and headed for the same fate. Whether the vision Kosh is somehow "of Kosh" is unknown, but I tend to think that it is. I'm also inclined to think that Kosh is becoming keenly aware of the nature of the struggle between the Vorlons and Shadows.
I don't think there is anything specifically affecting the Narn or Centauri like a disease or something. Rather that they are on a collision course and ultimately everyone is going to lose.
Kosh may even also be referring to the Vorlons and Shadows (as subtext). The Narn and Centauri, consumed by their hatred of each other, will destroy each other. Only if they let go of hate, can they ever hope to survive.
The wise, thoughtful G'kar we see in later seasons was always there,but his true self was buried beneath a sea of rage at what the Centauri had done to his world.
When Kosh says they are a dying race and we should let them pass. He is referring to dying in the literal sense. They are dying off. Centauri space is a fraction of what it was before. Nothing more than a trinket producing society embroiled in its own politics and societal standing. Having made little progress in the last hundred or so years, since they met the Narn.
The Narn, since meeting the Centauri. Have changed. They care only about revenge, what is right, meting out justice. Their sole drive is to kill the Centauri. They don't engage in things unless they can help meet this purpose.
The two are locked in a battle. But not one like the Shadows and the Vorlon. That is a battle of ideologies. The Centauri and Narn are locked in a battle to the death. They will kill each other. Until only one Centauri and one Narn remain and even they will still fight.
When he says "let them pass" he means let them die off. Let them fight amongst themselves until there is nothing left. Same as you say someone "passed" onto the next life. The Narn and Centauri will pass on.
What he is really saying is that unlike the Humans and the Minbari, who will and do reach the same "level" as the Vorlon in terms of society, technology, progress, ideals, etc. The Narn and Centauri never will. They will die off before they can reach that state. He simply sees that before anyone else does. As he has likely seen it hundreds of times before.
It could also be in regards to telepaths. The Narn telepaths were wiped out during and just after the first Shadow War, and as teeps are a evolutionary next step, once they are all gone, the race won't evolve any further.
With regard to the Centuri, (who at the time of the second Shadow War had telepaths) I think it was possible the Shadows were somehow killing off the Centuri Teeps in secret (although it was never in the show).
As the Vorlons created telepaths to use as weapons against the shadows in their wars (and because the Shadows wiped out teeps whenever they could) from the Vorlon perspective, a race without telepaths was as good as dead!
The Narn and the Centauri are dying because they are touched by the Shadows. Molari, Lord Rifa, and other Centauri no doubt, are imbued with the view of the Shadows, acting on what they want, dreaming of the return of military power and conquest, i.e. a struggle as the Shadows want all to act. The Narns, who earlier drove the Shadows off their homeworld apparently, are consumed by their revenge of the Centauri. Early on it is stated that the Narns have mistreated other races (unspecified)the way the Centauri had mistreated again. Again the Narns are acting on what they want. Kosh is stating the Vorlon view of any that are seeking personal agrandizement.
A dying race is a race that dies - a race that has not or is not capable of acheiving 'immortal' trandscendence. The Minbari & Humans will become like the first ones. The Narn & Centauri will continue dying, so, let them pass.