From the League wikia.

• Attacking structures does not activate the passive.
• Hitting a single enemy with an AoE spell does not proc the passive.
• DoT single target damage does not seem to proc the passive (needs further testing).
• Ranged champions making use of Runaan's Hurricane will still proc the bonus move speed from this enchantment.
• Pet's autoattacks do not proc the passive.

Dot Research:

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.

Trap Research:

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.

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.

From the League wikia.

• Attacking structures does not activate the passive.
• Hitting a single enemy with an AoE spell does not proc the passive.
• DoT single target damage does not seem to proc the passive (needs further testing).
• Ranged champions making use of Runaan's Hurricane will still proc the bonus move speed from this enchantment.
• Pet's autoattacks do not proc the passive.

Dot Research:

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.

Trap Research:

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.

I have a partial answer for you question based on the notes from the League wikia. I'll try to do some further testing tonight as far as the DoT effects and what traps are considered.

• Attacking structures does not activate the passive.
• Hitting a single enemy with an AoE spell does not proc the passive.
• DoT single target damage does not seem to proc the passive (needs further testing).
• Ranged champions making use of Runaan's Hurricane will still proc the bonus move speed from this enchantment.
• Pet's autoattacks do not proc the passive.

From the League wikia.

• Attacking structures does not activate the passive.
• Hitting a single enemy with an AoE spell does not proc the passive.
• DoT single target damage does not seem to proc the passive (needs further testing).
• Ranged champions making use of Runaan's Hurricane will still proc the bonus move speed from this enchantment.
• Pet's autoattacks do not proc the passive.

Dot Research:

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.

Trap Research:

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.

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.