I think I just came across an example of this.
I’m listening to an audio book by Kerry native speaker Sláine Ní Chathalláin called “Nain”.
And in the chapter “Muintir Nain” at around 4:30 she says “ní dh’fhág Nain talamh na hÉireann riamh”.
I thought this was quite interesting as the usage of dh here would only show up in the new past tense system (where all the r forms are dropped), which Sláine uses.

Séamus - by your original message, did you mean that forms like ní dh’fhágfad and ní dh’fhágann do not exist?

I have encountered dh' being used in the past tense alongside present negative particle (i.e. like ní dh'fhág, as you provided), but I have not yet found dh' being used in the present or future tense.

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I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

Ok so I've come across another usage of dh' - it seems like it's common (at least in Kerry and Ring) to use dh' in verb forms outside of relatives!
From listening through to the audiobook of Nain by Sláine Ní Chathalláin I'm coming across forms like dh'fhágadar, dh'ith sí, dh'éirigh sé etc.

Some questions if anyone knows the answer:
- I assume that the above also occurs in the other tenses where we use 'do'? i.e. past habitual and conditional?
- Would the above apply in Cork as well as Kerry and Ring?
- Would preference be given to the dh' forms vs the d' forms? As in, would the d' forms be considered unusual by comparison?

Given that dh' forms seem to be everywhere - do we ever see them in other parts of grammar, like inifinitive forms or the like e.g. "caithfir rud a dh'ithe?"
I assume no, as I thought that 'dh' was only just a drop in form for 'do'. But Seamus made a point about trying to avoid a hiatus in speech as much as possible, so I thought best to double check.

I think dh'éirigh is fine in Cork after eg a relative ( a dh'éirigh), but not "do dh'éirigh" that you find in Kerry literature.

rud a dh'ithe is fine too - see sar a' bhféadfaidís an fheóil a dh'ithe in Peadar Ua Laoghaire's Irish.
rud a dh'fháil is fine.

As for d' versus a dh', é d'fhágáil is found in some of Ua Laoghaire's books, and Donncha Ó Céileachair wrote that Muskerry natives would prefer é ' fhágaint there.