After a lot of messing around it seems that this is happening, regardless of wether I am wearing any shards/psalms, or not.

Tested plasma gun, plasma pistol, lasgun, laspistol, autogun, autopistol, shotgun, Meltagun, Inferno Pistol, Needler, below are results:

-I can apply DoTs with weapon skills that have it included in its description.

-Chance to apply DoTs on hit/crit are not applying regardless of wether they are weapon enchant, or a passive.

-Different combinations of passives, enchants did not make a difference.

-Unequipping and equipping gear, or restarting the game does not fix it.

-They do apply the DoT from the hit/crit proc once at the beginning of the mission but never apply it again, even after mission completion, and starting a new one. 

This makes my dual plasma pistol build so weak, because I cant stack heat DoT :( 

I restarted the game and was able to apply DoTs on the first couple of enemies, but then it just stopped applying entirely 

So I removed the psalm doctrine that gives the "+50% heat dmg and resistance is halved" from my chest armor, and my weapons started applying DoTs again. 

I had the issue a while back but since I switched from heat to physical for some time, I forgot about it. But it seems that equipping that psalm doctrine is stopping DoT effects from being applied. 

So guess that doctrine was the culprit, or the total combination of psalms across my gear. I do not know if the  warp/physical doctrine equivalent has the same issue of DoTs not applying, Ill test it later.

Main Implant: +50 dmg and 30% supreme dodge in surprise

Armor: +50 heat dmg and heat resistance halved to enemies

inoculator: 10% of dot dmg gained as suppression

Belt: +10 crit hit strength/Uncreator Psalm

Weapon Right: +125% Heat dmg

Weapon left: 4x -5% cd shards

I also have the passive heat crits apply burn but it does not work. I can confirm I have the same issue with the 2handed plasma gun with +125% heat dmg, and 2x -5 cd shards. On any other heat dmg weapon it all works fine.