DoT, as status effects, are not affected by crit for their damages but their durations. They are affected by Might however.
Even if they come from weapon or spell, they are not affected by bonus damage to weapon or spells. Because their damages cant crit, bonus for crit damages are useless too for them.
Only generic damage bonus,PL bonus and Pen bonus (see in comments) can work, for exemple :
High Harbinger's Robes provide +10% damages (for self's DoT)
Thaos' Headdress give +5% damage vs flancked (self)
Crucible token , 15% vs designed enemy (self)
Xoti's Lantern (Threshing Aura), <20% damage (everyone)
Baubles of the Fin, +3% (everyone)
Shattered Vengeance (Shards of Woedica) up to 25% taken (multiple effects can be obtain with summon by the way)
(the list of items giving some PL bonus is below)
Maybe I forgot some other items, and some others are specialized :
Dire Talon : Tooth and Claw ( +10 cc and damage against beast) is universal
Aamiina Legacy : Slayer of Beasts (+10 Accuracy and damage against Beasts) is universal
Wicked Beast : Wild Heart (+15% damage against Beasts) is universal
Amaliorra : Ash and Dust (+10 acc against Vessels) is universal
Amaliorra : Innevitable Decay (+20% damage against Vessels) is universal
Whispers of Yenwood : Soul Cutter (+20% Damage and +15 Acc against Spirits) is universal
Bane Spirit (+20% damage against spirits) is universal (from the Spoil of Caed Nua Mod)
There are also some abilities and passives that can increase the DoT damages :
Takedown Combo (ranger) give +100 damage until a new direct attack that deal damage (for everyone's DoT)
Hunter's Fang (ranger) < to 20% vs the enemies race (self)
Sworn Enemy (and upgrades, paladin), +20% damage vs designed enemy (self )
Inspired Beacon (paladin), +40% damage (everyone )
Blooded (barbarian), +25% damage when bloodied (self )
Wilder Hunter (barbarian), +25% damage vs wilder (self )
Fighting Spirit (human), +15% damage when bloodied (self)
Death's Usher (death godlike), +20% damage vs near death (self)
Infamous Captain (Watcher) +5% damage (self)
Ascended (Ascendant) +3 cipher PL, +20% damage (self)
Resting Bonus : Bloody Mess (The Kraken Eye, if you choose violence for resolve The Better Man) +10% damage (self)
Cleansing Flame (SC priest or scrolls) +100% tick rate during the effect
All these bonus are working on current and further DoT, and are not anymore effectives when over (Blooded, Fighting Spirit, etc)
For PL bonus, there are many way to gain some PL, from subclasses, races (godlikes), weapons, trinckets, ect. Considering only Generic PL, I made previously a topic about this, that take into account the scaling rules :
https://forums.obsidian.net/topic/126420-prestige-enchantment-the-scaling-transcendant-suffering-overview/
For Keyworded PL, every bonus stack on top of generic PL but not from two same weapons (2 Sun and Moon allow +2PL, not 4, and 1 Sun and Moon + 1 Magran's Favor allow +4PL).
By the same way, Acute and Brilliant from items seems to stack 1 PL with "ordinary" acute and brilliant. I'm not really sure because, when these effects are stacking, in the Character menu the second inspiration mention only the name of the inspiration, not the bonus (like ordinary).
Lance of the Midwood Stag (Lord of the Forest) +2PL (generic) while under a plant status effect (self)
Wicked Beast (Wild Heart) +2PL for Beast KW
The Spine of Thicket Green (empowering Instincts)+3PL for Beast KW or (empowering Lifeblood) +2PL for Plant KW
Magran's Favor (Fiery/Blazing Core) +2PL for Fire KW
Sun and Moon (Celestial Attunement) +2PL for Fire KW (day)
The Weyc's Wand (after empowering something) +3 generic PL
Spider Silk Robe (Poison Master) +2 Poison KW PL (require BPM to work)
Mask of the Grotto Deep +2Poison KW PL (Require BPM to work)
Some pets give +1 Fire KW PL
I'm not speacking about other KW PL because I dont see any DoT as Wind, Tempest, Electricity, Ice KW.
Alchemy increase the poisons (consumables) effect as well do generic and poison KW PL bonus.
Arcana increase the scroll PL as well do generic and correspondant PL bonus.