I don't believe so.

Flat damage increase, like elemental damage for instance, would only apply to the upfront hit from a fireball or thunderstrike not the DoT. There is a stat that directly increases the ailment damage, it reads something like % increased X ailment damage, where X is fire, lightning, rend, etc., that is the stat you are looking for if you want to increase your ailment's potency.

If regular increases to damage also applied to ailments that would make those stats far too strong. They would double dip in effectiveness.

Actually it should, it wouldn't make it overpowered either. I believe it's bugged. All spell damage % and elemental damage % doesn't add to current spells or DoT damage. Although, flat fire spells # to # does affect the spells and DoT damage. Even the Ailment element damage increase on weapons isn't working correctly. I've checked the character stats swapping weapons out and it doesn't calculate correctly.

I did some testing on this earlier today. Damage type mods DO in fact scale ailment damage. Rend damage, toxic, lightning, fire. For instance %increased lightning damage will increase your damage from lightning ailments. So will the catch-all %damage modifier. What does NOT scale ailments is spell damage / attack damage. As, the ticks of the ailment are not coming from a spell or attack.

I hope you are aware that as of right now the key passive Primordial Insights appears to have no effect. So until there is a bugfix it seems impossible to scale ailment damage via crit damage.

Great info! I've been meaning to test this out myself but I my heuristic, which is often "don't bother until I need to" can be difficult to overcome! Genesis stone in weapon is now a must!

Oh man, I just noticed this earlier tonight when making my poison ailment build and I was wondering WTF was going on with that passive.

I had to play around with big boi mobs that have lots of HP and I went into setting to turn on and off reg white dmg and yellow crit dmg numbers on the screen.
With just non-crit dmg turned on I could rapidly see the white #'s flash on the screen with my dots (stopping a few sec after casting and just watching). Note that I had the passive for sure selected so when I turned off reg dmg and just turned on crit dmg I saw no #'s flashing on my screen (and I had 50% crit total). The mobs just kept dying from what I presumed was normal ailnment dmg.
The only time I saw the yellow crit numbers was when I was channeling the spell on them initially...so safe to say I respecced out of that. Super annoying.
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As for OP's question, yes, flat +% toxic dmg, lightning, fire, rend or flat sub-category Elemental or Material or Occult will add to ailment dmg (is this current bugged too? I'm not sure).
There are a couple nodes on the GoF that give flat +8% dmg or something and those are global too.

Someone already pointed out that if the modifier is present (+spell/+attack) dmg then that will only affect the initial direct dmg.
Same with gems that say "+poison dmg added to spells or attacks". Those have a different socket that usually comes with "+poison ailment" or "+lightning ailment" out of the (I)(II)(III) socket options. So look for that on gear too.
I've only found two staffs so far that had +burn dmg and +poison ailment or whatever.