In League of Legends if I had a Rylai's Crystal Scepter (which has slow on spell hits as its passive) and I autoattack as Teemo, who has a magic DoT poison on hit, would it apply the passive slow from Rylai's Crystal Scepter?
No. Teemo's toxic shot cannot proc Rylai's Crystal Scepter, as it is an on-hit effect. In general, on-hit effects do not trigger item abilities or spell vamp.
Other DoTs (which are not on-hit effects), however, do trigger the Rylai's slow, like Mordekaiser's Ultimate, and Malzahar's Malefic Visions.