Can I use Darkness to have a chance to dodge damage from a damage over time tick or other similar effects?

Solution 1:

On the Beta forums (which are now deleted, here's an archived link to the thread), a developer, Celestalon, responded to this question in decent detail. I've added emphasis to the most important bit.

Each affected player individually gets a +20% chance to be missed by melee, ranged, and spell attacks. Some encounter mechanics may bypass this if needed to ensure the encounter still functions. The intent is for it to work as a different form of damage reduction for a group, such as against Megaera's intermissions. It's not intended to give you a random chance to avoid phases or major mechanics, such as Mannoroth's Doom Lords.

This is to say, all incoming attacks (melee, ranged, and spell attacks) are given a 20% chance to be ignored. Damage over time effects are not mitigated by this whatsoever, so this is useful when your group is about to be hit by multiple individual attacks from various sources, or to give your tanks a chance to completely ignore melee attacks by a boss. It is not designed to simply reduce damage, but rather to offer a chance for certain attacks to be ignored. Area of effect spells can be ignored, though!

For a current example of when Darkness could be useful, the Begrudging Waiters, spawned during the fight against the Council of Blood in Castle Nathria, attack members of your raid group at random. Grouping your raid members together and casting Darkness will give each attack a chance to be ignored. Considering the waiters attack relatively quickly, you should be able to ignore a surprising amount of damage over the course of the Darkness buff. Your healers will thank you!