Christianity believes that God takes human shape in the form of Jesus, and comes down to earth as a man. Of course, Jews do not believe that God ever takes the shape of a human, but how do we know this? We are commanded not to make a graven image or likeness of God, but how do we know that God does not create a human likeness of himself?
The Rambam considered such an idea a violation of the Unity of God (Perush haMishnah, Introduction to Perek Chelek):
This is to accept that this Oneness that we have mentioned above (Principle II) is not a body and has no strength in the body, and has no shape or image or relationship to a body or parts thereof... And if He were to be a body then He would be like any other body and would not be God... This is the third pillar and is attested to by the verse, “For you saw no image” meaning that you did not see an image or any form when you stood at Sinai because as we have just said, He has no body, nor power of the body.
This is also a simple understanding of the passuk the Rambam quotes. Do not create any image or form of God since you saw no image, i.e. God does not take any physical form, so it would be a terrible thing to create such a false image.
טו ונשמרתם מאוד, לנפשותיכם: כי לא ראיתם, כל-תמונה, ביום דיבר י'ה'וה אליכם בחורב, מתוך האש. טז פן-תשחיתון--ועשיתם לכם פסל, תמונת כל-סמל: תבנית זכר, או נקבה ...
Fine answer by Ariel K... Here's some more sources to supplement it:
1) Malachi 3:6 : "I am the Lord, I have not changed...". That God hasn't changed also implies the impossibility of taking on physical form, which would necessarily imply change. (Of course, the human body, indeed all physical matter are in the state of constant flux, which also precludes the possibility of immutability.)
2) Numbers 23:19: God is not a man...
That which is Eternal, by definition, cannot be preceded by anything. Hence, the Eternal cannot be preceded by any framework of existence.
G-d does not exist inside some framework of existence, like our bodies exist in spacetime. Rather, He is the existence and also the framework of existence of everything. Both things simultaneously. Hence there cannot be anything or anywhere devoid of Him. A body implies some kind of "place" where He is and another place where He is not and this is impossible in light of the above. source shaar yichud of chovos halevavos as I understand it.
Before my answer, note that whether G-d has a body or not has already been discussed. I understand this question to be discussing something different: G-d providing a vision where He seems to assume human form (not that it is His actual form, it's just a manifestation, similar to the shechina or ananei hakavod). You assume G-d doesn't and want to know why can't it happen. However, we see a few examples where Chazal seem to describe Hashem create a human manifestation.
תניא א"ר ישמעאל בן אלישע פעם אחת נכנסתי להקטיר קטורת לפני ולפנים וראיתי אכתריאל יה ה' צבאות שהוא יושב על כסא רם ונשא
Similarly, it was taught in a baraita that Rabbi Yishmael ben Elisha, the High Priest, said: Once, on Yom Kippur, I entered the innermost sanctum, the Holy of Holies, to offer incense, and in a vision I saw Akatriel Ya, the Lord of Hosts, (one of the names of God expressing His ultimate authority), seated upon a high and exalted throne.
There seemed to be some manifestation of G-d which was sitting that R' Yishmael saw.
Another example in Rosh Hashanah 17b
ויעבור ה' על פניו ויקרא א"ר יוחנן אלמלא מקרא כתוב אי אפשר לאומרו מלמד שנתעטף הקב"ה כשליח צבור והראה לו למשה סדר תפלה
The verse states: “And the Lord passed by before him, and proclaimed” (Exodus 34:6). Rabbi Yoḥanan said: Were it not explicitly written in the verse, it would be impossible to say this, as it would be insulting to God’s honor. The verse teaches that the Holy One, Blessed be He, wrapped Himself in a prayer shawl like a prayer leader and showed Moses the structure of the order of the prayer.
Now this answer could depend on how literal to take these passages (see this article for a discussion on how literal to take midrashim), but I don't see why G-d can't create a vision of a person for some prophetic experience just like he creates a vision of a cloud to represent His presence (Exodus 16:10).
While there are other discussions about this on the exchange, HaShem does take some type of form when He speaks to Moses.
Nubmers 12:8 - With him I speak mouth to mouth; in a vision and not in riddles, and he beholds the image of the Lord. So why were you not afraid to speak against My servant Moses ?
The very word used for image is the very same word in Exodus 20:4 and in Deut. 4:12 where it says do not make an image of HaShem.
Therefore, while we cannot make an image of HaShem, He can take an image, at least per Numbers 12:8. There are other manifestations of HaShem throughout the Torah and the Prophets as well.
Also note, that the ark, sukkah, and the tabernacle/Temple all are an image of HaShem. So it is almost like we are not to make an image of Him unless He tells us to make an image of Him.