It is very clear that Allah is not everywhere rather He is above the heavens and the earth, beyond the creation/not within the creation (Hadith in Muslim). So I wonder, where did the concept of Allah being everywhere come from? And when did it begin?
This is one of the main implications or principles of Wahdat Al-Wujud. Basically means every creature that the eye sees is Allah in essence. This concept is held by several so called soofee saints like Ibn Al-Faarid, Ibn ‘Arabi, Ibn ‘Sab’een, Al-‘Afeef At-Tilmisaani, Ibn Masheesh, An- Nabulusi and others. Many Soofis consider Ibn ‘Arabi as the symbol of the concept of Wahdat Al-Wujud.
Ibn 'Arabi considers the Soofee who knows of Allaah (‘Aarif billaah) as the one who “Sees Allaah in everything, rather he sees Him the essence of Everything," and that “He (Allaah) is the Essence of Existence” [Ibn ‘Arabi’s Fusoos al-Hikam [Beirut, Lebanon: Daar Al-Kitaab Al-‘Arabi, 1400 AH/1980 C.E.], p. 192.]
Historically, it started with Jahmiyyah and Mu'tazila who took their principles from Aristotle and Democritus.
Source: Sufism- Origin and Development
This is a question that has kept theologians very busy over the centuries. There are literally entire books about this one issue from opposing camps trying to prove their side.
Very briefly though, this issue arises because of a difference of opinion on epistemology: "How do we know what is true?" One camp (the kalamis and Ash'ari theologians) argue that revelation must be interpreted through reason, a specific branch of reasoning derived from Aristotelian logic, to be precise. To be extremely simplistic, one path their argument takes in this particular issue is as follows: Ascribing a direction to Allah implies that He occupies space, implying that He needs space to exist, implying that He is in need of something, whereas the Qur'an says He is Al-Ghani (free of need from anything), and therefore He cannot have a direction and He must be everywhere. There are a few other similar paths of reasoning but they all end up at the same place. Now the problem is to deal with relatively clearcut ahadith and athaar that contradict conclusions derived in this framework - they are either dismissed as abrogated, misunderstood, or in some cases, dismissed as simple-minded sayings from people who didn't understand how things really worked.
The position of the other camp is relatively simple - they accept the ahadith and athaar as they have reached us without ascribing a "how" to them (i.e. without taking them literally as we understand it). They take everything through the ayaat of affirmation (re. Allah's Attributes and ayaat in the Qur'an) and negation (e.g. the ayah that there is nothing like unto him) and go no further.
Source: 'Aqeedah class with Shaykh Yasir Qadhi, and other classes with other teachers
Neither Allah is everywhere in the sense that He has occupied the whole space or share position with everything, nor is He above the heavens and the rest, he is location-less, he is the Creator, the creator of space and time, so beyond all such notions. "Allah is everywhere" but is a correct statement, not in its apparent meaning but it needs a deep insight to be fully understood.
أَوَّلُ الدِّينِ مَعْرِفَتُهُ وَ كَمَالُ مَعْرِفَتِهِ التَّصْدِيقُ بِهِ وَ كَمَالُ التَّصْدِيقِ بِهِ تَوْحِيدُهُ وَ كَمَالُ تَوْحِيدِهِ الْإِخْلَاصُ لَهُ وَ كَمَالُ الْإِخْلَاصِ لَهُ نَفْيُ الصِّفَاتِ عَنْهُ لِشَهَادَةِ كُلِّ صِفَةٍ أَنَّهَا غَيْرُ الْمَوْصُوفِ وَ شَهَادَةِ كُلِّ مَوْصُوفٍ أَنَّهُ غَيْرُ الصِّفَةِ فَمَنْ وَصَفَ اللَّهَ سُبْحَانَهُ فَقَدْ قَرَنَهُ وَ مَنْ قَرَنَهُ فَقَدْ ثَنَّاهُ وَ مَنْ ثَنَّاهُ فَقَدْ جَزَّأَهُ وَ مَنْ جَزَّأَهُ فَقَدْ جَهِلَهُ وَ مَنْ أَشَارَ إِلَيْهِ فَقَدْ حَدَّهُ وَ مَنْ حَدَّهُ فَقَدْ عَدَّهُ وَ مَنْ قَالَ فِيمَ فَقَدْ ضَمَّنَهُ وَ مَنْ قَالَ عَلَامَ فَقَدْ أَخْلَى مِنْهُ كَائِنٌ لَا عَنْ حَدَثٍ مَوْجُودٌ لَا عَنْ عَدَمٍ مَعَ كُلِّ شَيْءٍ لَا بِمُقَارَنَةٍ وَ غَيْرُ كُلِّ شَيْءٍ لَا بِمُزَايَلَةٍ فَاعِلٌ لَا بِمَعْنَى الْحَرَكَاتِ وَ الْآلَةِ بَصِيرٌ إِذْ لَا مَنْظُورَ إِلَيْهِ مِنْ خَلْقِهِ مُتَوَحِّدٌ إِذْ لَا سَكَنَ يَسْتَأْنِسُ بِهِ وَ لَا يَسْتَوْحِشُ لِفَقْدِهِ
The foremost in religion is the acknowledgement of Him, the perfection of acknowledging Him is to testify Him, the perfection of testifying Him is to believe in His Oneness, the perfection of believing in His Oneness is to regard Him Pure, and the perfection of His purity is to deny Him attributes, because every attribute is a proof that it is different from that to which it is attributed and everything to which something is attributed is different from the attribute. Thus whoever attaches attributes to Allah recognises His like, and who recognises His like regards Him two; and who regards Him two recognises parts for Him; and who recognises parts for Him mistook Him; and who mistook Him pointed at Him; and who pointed at Him admitted limitations for Him; and who admitted limitations for Him numbered Him. Whoever said in what is He, held that He is contained; and whoever said on what is He held He is not on something else. He is a Being but not through phenomenon of coming into being. He exists but not from non-existence. He is with everything but not in physical nearness. He is different from everything but not in physical separation. He acts but without connotation of movements and instruments. He sees even when there is none to be looked at from among His creation. He is only One, such that there is none with whom He may keep company or whom He may miss in his absence.
being everywhere implies both being aware of everything everywhere (how doesn't He know when He Himself has created everything instant by instant? --according to Imam Sadiq peace be upon him) and that everything is a sign of His being, His existence, His uniqueness, His mercy, His wisdom, and etc. . You read a book and partly understand its writer, the writer is not necessarily talking about himself clarifying how he is but you can know him better, and better if you read more of him. We see the universe God has created and we understand His attributes better and better, until we would say:
... رَبَّنَا مَا خَلَقْتَ هَٰذَا بَاطِلًا سُبْحَانَكَ فَقِنَا عَذَابَ النَّارِ
... "Our Lord, You did not create this aimlessly; exalted are You [above such a thing]; then protect us from the punishment of the Fire. [3:191]
As answers above cite secondary theology, I would like to remind a passage from the Holy Quran:
God! There is no god except He, the Living, the Everlasting. Neither slumber overtakes Him, nor sleep. To Him belongs everything in the heavens and everything on earth. Who is he that can intercede with Him except with His permission? He knows what is before them, and what is behind them; and they cannot grasp any of His knowledge, except as He wills. His Throne extends over the heavens and the earth, and their preservation does not burden Him. He is the Most High, the Great.
«His Throne extends over the heavens and the earth» I interprete it in the sense that God is actually everywhere, omnipresent. HE is inside us, in evry worm, microbe, atom or quark, and He extends over all cosmos.
The cosmology of heavens has been developed in various «revelations», mostly a kind or religoius fiction, before Hadith, and most Hadith retell prior tellings, even if attributed to the Prophet ﷺ. Little of this cosmology is to be found in recognised revelation or the Quran, Gospels or Torah.
I found interesting aspects of cosmology in Al-Mulk (Surah 67) In particular 67:5
We have adorned the lower heaven with lanterns, and made them missiles against the devils; and We have prepared for them the punishment of the Blaze.
The lanterns are the stars we see; the whole cosmos we know. With todays knowledge, imagine what it means stating that there are (as mentioned in 67:3, 2:29, 17:44, 65:12, 71:15) seven heavens ...