TLDR
- I have not been able to find anything else regarding this list. With more books coming out maybe we will get an update on some of the abilities that each primarch has. I still was not able to find anything definitive for only Dorn and A&O. Of the other 16 primarchs I was able to find sources for 7 of the claimed abilities. Of the other 9 claims I have heard of them 4 or 5 of them previously. As I have mentioned before however, I subscribe to the idea that all of them have some abilities, even if they were latent and undiscovered.
- (Thanks to @T.J.L for cleaning this up after my original post...)
Regarding my list, the ones I have marked as "Yes" all have their source behind their described abilities. Still that is almost half of the known primarchs wholly confirmed. With the rest of the list only missing 2 with identified abilities this would mean that of the 20 primarchs, 16 are confirmed to have had some ability. As has been mentioned before, just like the Emperor and the Astartes, the primarchs are not human.
As I already mentioned, I subscribe to the philosophy that all of the Primarchs were psykers to some (likely a great) extent. While these abilities may not have been planned by the Emperor and developed by either the Chaos Gods throwing them across the galaxy or the planets they grew up on (Why I would argue Magnus became so powerful) there is still one key aspect to remember.
All psykers have a strong presence in the warp... And given the primarchs are all mini-emperahs, they definitely have a significant presence in the warp (I believe this is occasionally covered when Magnus or Ahriman are doing their "warp surfing" thing).
In addition, from one of the Space Wolves codices - "...the warriors of the twenty Space Marine Legions echoed to some degree the particular strengths and powers of the Primarch whose genes were used to develop their own..."
Even if we can not find/prove an example of a psyker trait for all 18 of the known primarchs it is possible that, like some psykers even, the ability remained dormant/latent and they were never really aware of their abilities.
Another great point I read online mentioned how these latent abilities might have even been the sources of their identifying characteristics, like the Lion's strategic cunning for example. Maybe his ability to gauge the flows of the battlefield and see the weaknesses of his enemy's were partially driven from some persistent psyker abilities. He never looked closer into why he was so good at his job, and just accepted the fact that he was a BAMF...
THE LIST
Lion - Yes, sense daemons and could see them when they're invisible to non psykers. He also had a unique ability to charge his weapon with psychic energy, he basically casts force on it; reading minds while protecting his own? (Angels of Caliban)
Fulgrim - Yes, psyker shield that he lowered before being possessed by a daemon / Fulgrim figures out his psychic power and cripples a Warhound Titan with it (A Reflection Crack'd)
Perturabo - Maybe, Could sense Warp taint and could see the Eye Of Terror no matter where he was
Khan - Maybe, Khan can kill psychanuein with his sword even though it's supposedly just a regular power weapon (as regular as a Primarch's weapon can be). Normally Psychanuein can only die to psychic attacks (Brotherhood of the Storm)
Russ - Yes, battle against thousand sons (A Thousand Sons)
Dorn - ?
Curze - Maybe, foresight to worst outcome and "bending shadows"
Sanguinius - Yes, death imprinted his legion and foresight (Corbulos-Codex)
Ferrus - Maybe, psychic storm when he died and he was able to destroy weapons by touching them? / Necrodermis forearms
Angron - Maybe, had some psychic chat with Sanguinius at some point? / loves to cast his skull-crushing magic with his chainwand.
Roboute - Maybe, able to detect psychic energy in a way that a true non-psyker wouldn't be capable of. Guilliman acknowledges that the sword actually catching fire means he must be a bit of a psyker, though he doesn't want to push the envelope with it.
Mortarion - Maybe, was exceptionally fast in learning how to cast sorcery and incredibly resident to poison, even by Primarch standards
Magnus - Yes, etc. (Pick a book)
Horus - Maybe, during final battle (though I guess that could have been from Chaos influence)
Lorgar - Yes, visions of the Emperor's arrival on Colchis (Aurelian)
Vulkan - Maybe, Perpetual immortality?
Corvus - Yes, invisibility through mind trickery (Deliverance Lost)
A&O - Maybe, a unexplained mental understanding with each other. Not sure if telepathy, twin familiarity or genuinely one soul in two bodies.