Unique Passive: Dealing spell damage applies a damage-over-time effect that deals bonus magic damage equal to 2% of the target's current health per second for 3 seconds. This bonus damage is doubled against movement-impaired units and capped at 100 damage per second vs. monsters.
Rylai's Crystal Scepter will slow champions on-spell-hit but the slow only lasts 1.5 seconds while the DoT occurs over 3 seconds, so only first half of DoT will be doubled.
Deathfire Grasp deals percent magic damage based on maximum health, while Liandry's Torment deals percent magic damage based on current health, which might intuitively seem to not synergize well together. However, because Deathfire Grasp's damage is reduced by the enemy's magic resist, its 15% damage amplification will still result in an overall increase to Liandry's Torment's damage. Liandry's Torment will only have its maximum damage contribution mitigated if Deathfire Grasp's active is used when the opponent's current % health is below 100% - % damage reduction from resists and other factors. For example, if an opponent has an effective 40% magic damage reduction, it is efficient to use Deathfire Grasp's active while the opponent's current health is above 60% (100% - 40% = 60%). Ultimately, because Liandry's Torment is based on current health, its contribution becomes smaller as the opponent loses health; therefore, using Deathfire Grasp's active on an opponent with a % current health lower than this rule still generally results in a very small loss in efficiency. In practical application, any potential anti-synergy between the two items can be disregarded as negligible.
Unique Passive: Dealing spell damage burns enemies for 5% of their current health as magic damage over 3 seconds. If their movement is impaired, they take double damage from this effect. Duration is halved for multi-target or periodic effects, 300 max damage vs monsters.