Would Ilúvatar raise Númenor if King Elessar and the Gondorians asked him to, or is Númenor gone forever and no way to get it back?
Answers that relate to Tolkien's own writings are strongly preferred.
The short version is sort of. He won't if they asked, but he might eventually. Let's talk about why he won't, first. There are really three reasons.
In "Appendix B: The Tale of Years", Tolkien lists some significant events that took place during the Fourth Age, up to Aragorn's death in FA 120 (1541 by Shire Reckoning). The rising of Numenor would surely count as a noteworthy event, but no mention is made of it.
On a related note, Tolkien intended that the downfall of Numenor be a version of the classic Atlantis myth. Even if the narrative similarities don't convince you (a noble island civilization drowned by the gods because of the hubris of their leaders), we have a tacit admission from Akallabêth:
And even the name of that land perished, and Men spoke thereafter not of Elenna, nor of Andor the Gift that was taken away, nor of Númenórë on the confines of the world; but the exiles on the shores of the sea, if they turned towards the West in the desire of their hearts, spoke of Mar-nu-Falmar that was whelmed in the waves, Akallabêth the Downfallen, Atlantë in the Eldarin tongue.
The Silmarillion IV Akallabêth
And the man himself confirmed the link of some of his letters; in Letter 154:
The particular 'myth' which lies behind this tale1, and the mood both of Men and Elves at this time, is the Downfall of Númenor: a special variety of the Atlantis tradition.
The Letters of J.R.R. Tolkien 154: To Naomi Mitchison. September 1954
And Letter 156:
So ended Númenor-Atlantis
The Letters of J.R.R. Tolkien 156: To Robert Murray, SJ (Draft). November 1954
And Letter 227 (emphasis his):
The legends of Númenórë are only in the background of The Lord of the Rings, though (of course) they were written first, and are only summarised in Appendix A. They are my own use for my own purposes of the Atlantis legend
The Letters of J.R.R. Tolkien 227: To Mrs. E.C. Ossen Drijver (Excerpt). January 1961
It's not entirely clear that mortals are capable of initiating a conversation with Eru from within Arda.
It is clear that Eru is capable of communicating with beings inside Arda if he so chooses. For example, in The Silmarillion he communicates with the Vala Aulë, creator of the Dwarves:
Ilúvatar spoke to him; and Aulë heard his voice and was silent. And the voice of Ilúvatar said to him: 'Why hast thou done this? Why dost thou attempt a thing which thou knowest is beyond thy power and thy authority? For thou hast from me as a gift thy own being only, and no more; and therefore the creatures of thy hand and mind can live only by that being, moving when thou thinkest to move them, and if thy thought be elsewhere, standing idle. Is that thy desire?'
Then Aulë answered: 'I did not desire such lordship. I desired things other than I am, to love and to teach them, so that they too might perceive the beauty of Eä, which thou hast caused to be.'
The Silmarillion III Quenta Silmarillion Chapter 2: "Of Aulë and Yavanna"
And from Morgoth's Ring we have a bit of lore from the earliest days of men, called "The Tale of Adanel":
The Voice had spoken to us, and we had listened. The Voice said: 'Ye are my children. I have sent you to dwell here. In time ye will inherit all this Earth, but first ye must be children and learn. Call on me and I shall hear; for I am watching over you.'
In that time we called often and the Voice answered. But it seldom answered our questions, saying only: 'First seek to find the answer for yourselves. For ye will have joy in the finding, and so grow from childhood and become wise. Do not seek to leave childhood before your time.'
[Morgoth comes to Men, and they worship him rather than Eru]
The first Voice we never heard again, save once. In the stillness of the night It spoke, saying: 'Ye have abjured Me, but ye remain Mine. I gave you life. Now it shall be shortened, and each of you in a little while shall come to Me, to learn who is your Lord: the one ye worship, or I who made him.'
Then our terror of the Dark was increased; for we believed that the Voice was of the Darkness behind the stars. And some of us began to die in horror and anguish, fearing to go out into the Dark. Then we called on our Master [Morgoth] to save us from death, and he did not answer.
The History of Middle-earth X Morgoth's Ring Part 4: "Athrabeth Finrod ah Andreth"
It's strongly implied that The Voice is Eru, and in fairness it does rather sound like him. I'm hesitant to take this at face value though, for two reasons:
For one thing, the lore comes from a human woman relating an oral history that's been passed down for about 400 years. Tolkien often reminds us in his letters that the characters in his books are not all-knowing, so it's entirely possible that the story has been corrupted somewhat.
Secondly, that bit at the end about death seems to contradict later writings. It reads as though early men were immortal (And in an earlier story from Morgoth's Ring, "Athrabeth Finrod ah Andreth", it's stated that this is the belief of the current generation), and Eru took immortality from them in punishment for worshipping Morgoth.
Now it's possible that all this took place within the first generation of men, and they just had no experience with death because they'd only been alive for a few decades by now, but it does seem like rather convenient timing for that.
The contradiction comes from early in The Silmarillion:
It is one with this gift of freedom [death] that the children of Men dwell only a short space in the world alive, and are not bound to it, and depart soon whither the Elves know not. [...] But the sons of Men die indeed, and leave the world; wherefore they are called the Guests, or the Strangers. Death is their fate, the gift of Ilúvatar, which as Time wears even the Powers [The Valar] shall envy.
The Silmarillion III Quenta Silmarillion Chapter 1: "Of the Beginning of Days"
The intention here is that men were mortal from the get-go, which seems to contradict the passage from Morgoth's Ring.
Even if we can take "The Tale of Adanel" at face value, it sure reads like Eru is closing off lines of communication. Whether he's even listening any more, or if he'd respond, is unknown.
There's also evidence that, men are still trying to contact Eru, even thousands of years later. In Akallabeth we have the following description:
[I]n the midst of the land was a mountain tall and steep, and it was named the Meneltarma, the Pillar of Heaven, and upon it was a high place that was hallowed to Eru Ilúvatar
The Silmarillion IV Akallabêth
Before they turned to Morgoth, the King of Númenor would say prayers to Eru from the top of Meneltarma. So clearly they're trying to speak to him, although they probably don't intend to communicate in the same way they did in the earliest days. It's also unclear whether not not Eru ever responded, though I doubt he did; I like to think that the Kings wouldn't have turned to Morgoth had they received some sign from Eru. But we don't know.
Eru really, really doesn't like getting involved in the affairs of Arda. I'm not sure that his non-interference is explicitly mentioned in any writings, but we know that in all the time since the world was created - over 20 000 years by the end of Return of the King - he's only interfered five times:
Each one of those was part of a spectacularly significant event. Aside from number 3, each one was a last-ditch effort to preserve the existence of Elves and Men, the Children of Eru. Number 3 is a punishment for the arrogance of Ar-Pharazôn, and should indicate just how great his transgression was that Eru would step in. Eru has a long memory (He's literally timeless), so it's unlikely that he'd be willing to rescind the punishment after a "mere" 3000 years. Even if he were so inclined, it's very doubtful that restoring Númenor would be a world-changing event that Arda literally could not survive without.
So, even if they could get him on the line, he wouldn't help them.
But, I said earlier that he is going to (Or he did, depending on how you look at it). Maybe.
In Return of the King chapter "Many Partings"2, there's a sequence where about a whole bunch of characters go visit Treebeard at the remains of Orthanc. At the end of this sequence, we get the following:
Then Treebeard said farewell to each of them in turn, and he bowed three times slowly and with great reverence to Celeborn and Galadrial. 'It is long, long since we met by stock or stone, A vanimar, vanimalion nosatari!3' he said. 'It is sad that we should meet only thus at the ending. For the world is changing: I feel it in the water, I feel it in the earth, and I smell it in the air. I do not think we shall meet again.'
And Celeborn said: 'I do not know, Eldest.' But Galadrial said: 'Not in Middle-Earth, nor until the lands that lie under the wave are lifted up again. Then in the willow-meads of Tasarinan we may meet in the Spring. Farewell.'
The Return of the King Book VI Chapter 6: "Many Partings"
The bold is my emphasis, and what I want to talk about.
The "lands that lie beneath the waves" could only be referring to two places: Beleriand, the main setting of The Silmarillion which was sunk during or after the War of Wrath at the end of the First Age of the Sun, and Númenor itself. "Tasarinan" is probably a reference to Nan-tasarinon, or Nan-tharen, which was a region in Beleriand that was known for its willow trees.
The bolded line as a whole may be a reference to the Second Prophecy of Mandos, which concerns the breaking and remaking of the world following Morgoth's final destruction. This prophecy was removed from The Silmarillion before publication, so its canonicity is unclear (Which is why I'm not quoting it here, it's also rather long). Regardless, clearly Galadriel at least believes that there will be a time in the future4 when Beleriand will be restored. It's possible that Eru will also restore Númenor at that point, but we just don't know.
1 For added context: the tale Tolkien refers to is the tale of how the Earth came to be round. It was originally flat, with Aman being, quite literally, the uttermost west of the world. As part of the downfall of Numenor, Eru curved Earth into a sphere and removed Aman from it.
2 This is the chapter that the infamous "dozen endings" of the Jacksonverse movie is based on. It features about a dozen of the named characters in the book travelling around in a pack, from Gondor to Rivendell, visting about another dozen characters along the way.
3 'O beautiful ones, parents of beautiful children!', according to Letter 230. A direct translation would be 'Fair ones, begetters of fair ones' according to what is by far the oddest thing I've ever read while researching for this site, What Tolkien Officially Said About Elf Sex
4 Possibly the distant future, since she's immortal and Treebeard is possibly the longest-lived mortal creature on Middle-Earth, if he's not immortal as well. This also indicates that Galadriel envisions returning to Arda at some point, which is somewhat surprising given that she returns to Aman only a few years after this chapter takes place.