What abilities proc the furor enchant on boots? Will stuff like Shaco's JitB auto attacks proc it? Will Nidalee's bushwhack give you the bonus movement speed each tic of damage? What about Caitlyn's yordle snap trap? And what if Karthus uses his requiem when there is only one surviving enemy player?
From the League wikia.
To test if DoT's proc furor boots I played a custom game versus Nasus Bot as Malzahar. After acquiring my furor boots, I cast my E (Malefic Visions) on him, which did not proc the furor enchantment. On a side note, I was also able to re-confirm that pets do not trigger the furor enchantment.
Edit: As Richard points out in the comments, Nidalee's traps are considered AoE so I re-tested Furor with Caitlyn Traps.
For the trap test, I played a game as Nidalee versus Trundle Bot. After acquiring my furor boots, I set a number of Bushwhacks and lured him into them. They did not proc the furor enchantment.
Example of the furor speed boost:
For reference you can see the Furor buff in the buff bar, and you can see the particle effects on Nidalee's feet.
Nidalee traps are considered AoE in that they can hit multiple targets if they are close enough when the trap is triggered. So, I re-tested this with Caitlyn and her cupcakes.
As you can see, they did not trigger the Furor enchantment on Caitlyn (see buff bar).
The most important thing to remember is Furor are considered on-hit effects, so for the most part you can use that as a guideline on what will or will not trigger their effects.