In order to answer the question, we must clarify the meaning of the infiniteness of God and His relation with other beings:
Divine infinity means that God is in every way immeasurable, unbounded, unlimited, unrestricted, without end, inexhaustible and has all attributes of perfection.[i] In all of His attributes, God is limitless and endless. When you consider His power, wisdom, justice, knowledge, etc., God is boundless. He is not restricted by time or space. He has all attributes of perfection and there is no perfection that He might lack.[ii] The essence of God is limitless and unbounded. The Quran says:
"ُوَ اللَّهُ فِی السَّماواتِ وَ فِی الْأَرْضِ"[iii] و " فَأَیْنَما تُوَلُّوا فَثَمَّ وَجْهُ اللَّهِ[iv]
"And He is Allah in the heavens and on earth" and "Whithersoever ye turn, there is the presence of Allah."
God is therefore present everywhere and He is not restricted by time or place. He is above time and place. However, infiniteness in relation to God does not mean that He has occupied all places in the same way and manner as the physical entities have filled them. In fact, God is not matter and He does not fill the places in the material sense. It is only matter (that has length, breadth and width) that occupies a place and God is not matter to occupy a place. Ayatollah Makarem Shirazi says:
"As for why God is not matter, it is because every material object is limited, limited by time, place, qualities, features and effects. Every physical object has a fixed dimension whereas God's existence is infinite and immeasurable in all terms. Obviously, such a being cannot be bounded with time and place.[v],[vi]
When it comes to the question as to why God is unlimited and infinite, we must say:
First, in view of the fact that a necessary being must be all-perfect (i.e. he should have all virtues of perfection) and that He cannot be limited by anything because if He were limited, it would mean that He is a combination of existence and non-existence. It has been proved else where that combination is a sign of need and the concomitant of need is space whereas God (Who is the Necessary Being) is free from need and imperfection as was argued above. In fact, combination is against our supposition.[vii]
Secondly, it has been proved in its place that God, the Exalted, is pure existence and pure actuality. Finiteness means combination of existence and non-existence. Therefore, pure existence is not compatible with finiteness and limitation, and pure perfection cannot be limited by anything.
[i] - Makarem Shirazi, Naser, The Belief of a Muslim, Hadaf Publication, 1st edition, pg. 38
[ii] - Makarem Shirazi, Naser, "How to Know God", Muhammadi Publications, 1343 pg.42
[iii] - Al-An'am, 3
[iv] - Al-Baqarah, 155
[v] - Makarem Shirazi, Naser, The Belief of a Muslims, pg. 43.
[vi] - Excerpted from answer question 1944 (site: 1944), Index: "God's Infinite Existence"
[vii] - Vide: A Collection of Student Questions, Khuda Shinasi and Farjamshinasi, Maaref Publication Office, 2003, pg. 71, with little modification.