How does an arahant know that ignorance won't return/re-arise after death?

Abhiññā knows Nibbana because abhiññā knows the defilements, including ignorance, no longer arise, in any situation, whatsoever.

The devas (mystics) with psychic powers who can read the mind of others also know the defilements do not arise in the mind of the arahant; even though those deva may not be arahants.

The Arahant

30. "This monk is called one who has removed the crossbar, has filled the moat, has broken the pillar, has unbolted (his mind); a Noble One who has taken down the flag, put down the burden, become unfettered.

31. "And how, monks, is that monk one who has removed the cross-bar? Herein the monk has abandoned ignorance, has cut it off at the root, removed it from its soil like a palmyra tree, brought it to utter extinction, incapable of arising again. Thus has he removed the cross-bar.

32. "And how, monks, is that monk one who has filled the moat? Herein the monk has abandoned the round of rebirths, leading to renewed existence; he has cut it off at the root, removed it from its soil like a palmyra tree, brought it to utter extinction, incapable of arising again.

33. "And how has he broken the pillar? He has abandoned craving, has cut it off at the root, removed it from its soil like a palmyra tree, brought it to utter extinction, incapable of arising again.

34. "And how has he unbolted (his mind)? He has abandoned the five lower fetters, has cut them off at the root, removed them from their soil like a palmyra tree, brought them to utter extinction, incapable of arising again.

35. "And how is the monk a Noble One who has taken down the flag, put down the burden, become unfettered? He has abandoned the conceit of self, has cut it off at the root, removed it from is soil like a palmyra tree, brought it to utter extinction, incapable of arising again. Thus is the monk a Noble One who has taken down the flag, put down the burden, become unfettered.

36. "When a monk's mind is thus freed, O monks, neither the gods with Indra, nor the gods with Brahma, nor the gods with the Lord of Creatures (Pajaapati), when searching will find on what the consciousness of one thus gone (tathaagata) is based. Why is that? One who has thus gone is no longer traceable here and now, so I say

MN 22

As for "death" ("marana"), this does not happen to Arahants, as written in many suttas. The word "death" ("marana") does not generally refer to the termination of life but refers to the "death" of a "self" or "a being" (which are forms of delusion).

You ask some interesting and reasonable questions.

Is it considered as being abhiññā i.e "direct knowledge" that occurs before the arahant dies?

I believe the answer is yes.

How does one know that this abhiññā is completely true? It seems to me that omniscience is required.

Omniscience is a difficult idea and there is much debate about what the Buddha's 'omniscience' really means. I would suggest omniscience is not necessary. What would be necessary, as a minimum, is a knowledge of the post-death state.

Doesn't truly knowing something like that require omniscience?

Logically-speaking omniscience would not be necessary, but it would require knowing more than most people would believe it is possible to know.

So much depends on the meaning of 'omniscience' and only an omniscient person could know.

Not good answers, I know, but I have to speculate. A minimum condition for the knowledge required would be direct knowledge of (bodily) death or 'what it is like' to be dead. Buddhism says this knowledge is available to us but there seems no reason to suppose this would in itself constitute omniscience.

In another discussion Rusi tells that when Sri Ramana Mahaharshi was asked about the pain from his cancer he concedes that there is bodily pain but notes 'I am not the body'. If we know we are not the body then we must know we can do without it.

Feel free to treat all this as idle speculation. It is a little more than this but cannot be called well-informed.

How it can be known that ignorance won't return/re-arise or why it won't return/re-arise if there is no consciousness at the time.

“Bhikkhus, there are these five higher fetters. What five? Lust for form, lust for the formless, conceit, restlessness, ignorance. These are the five higher fetters. The Noble Eightfold Path is to be developed for direct knowledge of these five higher fetters, for the full understanding of them, for their utter destruction, for their abandoning.

“What Noble Eightfold Path? Here, bhikkhus, a bhikkhu develops right view … right concentration, which has as its final goal the removal of lust, the removal of hatred, the removal of delusion … which has the Deathless as its ground, the Deathless as its destination, the Deathless as its final goal … which slants, slopes, and inclines towards Nibbāna. This Noble Eightfold Path is to be developed for direct knowledge of these five higher fetters, for the full understanding of them, for their utter destruction, for their abandoning.” SN 45.180

So ignorance won’t arise anymore because an arahant has cut off all 10 fetters. Ignorance is one of those fetters. An arahant have direct knowledge of these fetters and he knows their destruction and he also knows if the fetters are abandoned or not.

It seems to me that consciousness is required for knowledge, so when an arahant dies how can there be knowledge and thus certainty that ignorance won't reform/re-arise after death?

When an arahant dies he is free of the 10 fetters including ignorance and he also know the four noble truths so he will not be reborn. At the moment of his death, there was no ignorance so all things in the dependent origin that comes after ignorance will not arise because ignorance is not present at the moment of death.

Is it through direct knowledge of the entire future that occurs before death?

An arahant usually knows that birth is ended. It is usually mentioned in the suttas.

Is it known through abhijñā i.e "direct knowledge", of the entire future, before the arahant dies?

There was one sutta where a guy asked an arahant if he has psychic powers but the arahant replied no. I can’t find this sutta so can anyone tell me. So the abhijna or the six higher knowledge aren’t required to be an arahant except the last one which is the ending of mental fermentation. An arahant can develop psychic powers after becoming an arahant but they probably don’t have any desire to achieve those powers.

"With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, the monk directs and inclines it to the knowledge of the ending of the mental fermentations. He discerns, as it has come to be, that 'This is stress... This is the origination of stress... This is the cessation of stress... This is the way leading to the cessation of stress... These are mental fermentations... This is the origination of fermentations... This is the cessation of fermentations... This is the way leading to the cessation of fermentations.' His heart, thus knowing, thus seeing, is released from the fermentation of sensuality, the fermentation of becoming, the fermentation of ignorance. With release, there is the knowledge, 'Released.' He discerns that 'Birth is ended, the holy life fulfilled, the task done. There is nothing further for this world.' This, too, is how striving is fruitful, how exertion is fruitful. MN 101

So as you can see, an arahant knows that he is released and that there will be no further birth.

Doesn't truly knowing something like that require omniscience?

No, only the Buddha was omniscience. An arahant would know that there will be no further birth because

1) The Buddha told them that attaining arahantship will end rebirth

2) When he is released there is the knowledge “release”

3) He knows there is no next life by looking into the future with psychic powers.

I think 1 and 2 are the most possible way to know that he will not be reborn because not all arahants have psychic powers. It’s not even needed to look into the future because arahant knows that there is no further birth.

How does one know that the abhiññā is true? It seems to me that omniscience is required.

If you are literally flying then you would know there you are flying. If you can travel to other realms then you would know it. Omniscience is not required. You don’t need omniscience to know that your memories are real or not. If you try to remember what happened just an hour ago then you know that memory is true. In the same way, when you recollect your past lives you know that it’s true. Abhinna are as real as moving your hands or legs. So you will know if you have abinna or not.

How can it be known that ignorance won't return/re-arise and why won't it return/re-arise if there is no consciousness after death?

MN Atthakathā wrote:

The (mundane) vipassanā-right-view comprehends (all) aggregates of (all mundane) 3 realms by its 3 characteristics,

Vipassanāsammādiṭṭhi tebhūmakasaṅkhāre aniccādivasena parivīmaṃsati,

Realms (bhava) in pāli means birth, being, happening. 3 realms are all birthplaces. "comprehends (all) aggregates of (all mundane) 3 realms" means ceasing all ignorance and craving of birthplaces. No ignorance and craving are no birthplaces. No birthplace is no object. No object is no ignorance and craving, because ignorance and craving arise for ignoring and attaching their object. So, in MN Mahāsatipaṭṭhānasutta:

The taṇhā for visible forms in the world is pleasant and agreeable, there taṇhā, when abandoned, is abandoned, there when ceasing, it ceases. The taṇhā for sounds in the world is pleasant and agreeable, there taṇhā, when abandoned, is abandoned, there when ceasing, it ceases. The taṇhā for odors in the world is pleasant and agreeable, there taṇhā, when abandoned, is abandoned, there when ceasing, it ceases. The taṇhā for tastes in the world is pleasant and agreeable, there taṇhā, when abandoned, is abandoned, there when ceasing, it ceases. The taṇhā for bodily phenomena in the world is pleasant and agreeable, there taṇhā, when abandoned, is abandoned, there when ceasing, it ceases. The taṇhā for dhammas in the world is pleasant and agreeable, there taṇhā, when abandoned, is abandoned, there when ceasing, it ceases.

Next question:

---From what I remember seeing according to the linear view of the links of dependent origination ignorance is the cause of fabrications and fabrications are the cause of consciousness. ---It seems to me that consciousness is required for this type of knowledge, so when an arahant dies how can there be knowing and thus certainty that ignorance won't reform/re-arise after death if there isn't even consciousness?

The consciousness to know ignorance's cessation is required after enlightened as arahant imidiatelly, so it is not required after dead. Sutta. Saṃ. Ma. Bhikkhusutta:

While one is developing these seven factors of enlightenment, one’s mind is liberated from the taint of sensuality, from the taint of existence, from the taint of ignorance. When it is liberated there comes the knowledge: ‘It’s liberated.’ One understands: ‘Destroyed is birth, the holy life has been lived, what had to be done has been done, there is no more for this state of being.’ They lead to enlightenment, bhikkhu, therefore they are called factors of enlightenment.”

Next question:

Is it considered as being abhiññā i.e "direct knowledge" that occurs before the arahant dies?

It is called paccavekkhaṇañāṇa which appeared in above quoted Sutta. Saṃ. Ma. Bhikkhusutta, etc.

Ariya-magga stops avijjā&taṇhā in the future by stopping all present behavior of taṇhā&avijjā in every tense, past&future&present, because taṇhā&avijjā can't arise anymore without the present behavior in every tense. Paccavekkhaṇañāṇa can know the cessation of the future ignorance by checking that present behavior, so bhiññā is not required.

Next question:

How does one know that the abhiññā is true? It seems to me that omniscience is required.

I can know that the sun will rise at the east tomorrow without omniscience, and it is certainly true. So, if I die next minute, that future knowing still certainly true and it will be the future life of my present life. The omniscience and abhiññā are not required, right?

It's similar to the above example, but it is more power, greater.

Next question:

Doesn't truly knowing something like that require omniscience?

No, the omniscience is not required. It just makes the knowing clearer and easier to attain.