Does Jax's Grandmaster's Might passive trigger Deathfire Touch?

Jax's Grandmaster's Might passive does trigger Deathfire Touch.
I have not tested this myself (I might later and edit to confirm), but the wiki provides information as to why this is so.

On the Deathfire Touch page:

• Deathfire Touch is a spell effect and will only be applied to instances of ability damage.

Now we know Deathfire Touch is a spell effect, but does Grandmaster's Might passive count as ability damage? Yes.
On the spell effect page:

The following "on-hit effects" are incorrectly, although possibly intentionally classified as abilities and will trigger spell effects. The triggering attack can still trigger on-hit effects.
[...]

• Jax's Empower
  • Grandmaster's Might (passive)

And on the Jax page under Grandmaster's Might (click show ability details):

Grandmaster's Might is a self-buff ability that passively grants Jax an on-hit effect on all his basic attacks.

• Applies spell effects as a single target ability.
  • Spell vamp is fully applied.
  • Will of the Ancients will heal you for 15% of the raw damage dealt.
  • Rylai's Crystal Scepter will apply a 40% slow for 1.5 seconds.

Grandmaster's Might passive grants an on-hit effect of extra damage on every third basic attack in 2.5 seconds. Since this on-hit effect is classed as ability damage, it will apply spell effects and therefore Deathfire Touch.

I think it is safe to assume that, as @Gnarly 404 stated in the comments for @Eumel's answer, "[...] everything that triggers a Luden's Echo or Liandry's will also trigger Deathfire Touch." A quick look at some other abilities listed on the spell effect wiki page, such as Gragas's Drunken Rage and Kassadin's Nether Blade, confirms that this is probably the case. I did not check all of them, so feel free to prove me wrong.