What is the biblical basis for the claim that the Father is a bodiless spirit?

GOD is SPIRIT…HE doesn't 'have it' rather HE is IT. And SPIRIT has no limits (it can be/look like anything) so when HE is SPIRIT that isn't limited but rather unlimited.

Let me explain...

Paul shows us what SPIRIT form looks like. Part of the problem is that we can't see or really understand GOD or SPIRIT (same thing) with our visual (seeing) eyes.

Or as scripture says "by leaning on your own understanding" Proverbs 3:5

Paul tells us in 1 Corinthians 15 vs 40 and following, that everything has a different kind of body. People do, and animals do and birds and fish. Okay, that is simple enough.

But SPIRIT although invisible to our earthly eyes, also has its own heavenly body. Each (earthly body) and (Spiritual /Heavenly) body has a splendour and glory to it. They serve (these bodies) a purpose in the environment GOD puts us in.

JESUS said "the SPIRIT is willing but the flesh (earthly body) is weak." So, they have a splendour to them, and play a roll but the SPIRIT is superior without limitations (like our bodies have).

JESUS when HE rose, was in HIS glorified (SPIRIT) body. Not limited to what man (earthly body) is limited to. HE could appear here or there. HE looked radiant in fact some didn't recognize HIM (as HIS appearance was somewhat different). But HE still ate and loved and touched and could be touched (Thomas touching his hands and side). Also Mary holding onto HIM not letting go in John 20:17

So, GOD is SPIRIT it has a body but it is invisible to earthly flesh/man.

Here is the scripture of 1 Corinthians 15:39-42 I was discussing:

Not all flesh is the same: people have one kind of flesh, animals have another, birds another and fish another. There are also heavenly bodies and there are earthly bodies; but the splendor of the heavenly bodies is one kind, and the splendor of the earthly bodies is another. The sun has one kind of splendor, the moon another and the stars another; and star differs from star in splendour.

-ezeric

God is most assuredly a Spirit without any physical attributes.

Genesis 1:1 and 2 NKJV In the beginning God created the heavens and the earth. 2 The earth was without form, and void; and darkness was on the face of the deep. And the Spirit of God was hovering over the face of the waters.

These verses not only tell us that God is the creator of all things material, but that he existed in non-material form, before creation. So let's take a longer look at these Scriptures and see if we can better understand what they tell us.

Verse 1 is key to understanding not only creation, but also many other Scriptures such as John 4:24 cited by Victor Ng in his answer.

In the beginning God created the heavens and the earth. This brings up a question of what does in the beginning mean?

It is quite obvious that in the beginning is in reference to time.

The following is quoted from http://everythingforever.com/einstein.htm

Unfortunately it is very much a reality still today that the works of the really great spirits in science, such as Albert Einstein, Richard Feynman, and Stephen Hawking, and their most fundamental conclusions about the universe, remain unappreciated, even unnoticed by the majority of scientists. These three giants all made very similar conclusions regarding the timelessness of the Universe, yet even today the science of timelessness is overlooked.

Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.

The second statement comes from Einstein's theory of relativity. It is a common belief among these three individuals, (Albert Einstein, Richard Feynman, and Stephen Hawking)that time and motion are interchangeable, and Einstein once stated that if All motion within the Universe could be reversed, time itself would reverse.

Although it was not their intent this theory lends credence to three Omni aspects of God.

If time/motion does exist in a fourth dimension where all events are simultaneous then the Spirit would be able to be omniscient, omnipotent, and omnipresent; as an observer in that fourth dimension, and as it's creator would have the power to reverse time by reversing all motion within the Universe.

The more that Science learns about our environment, the more it validates God as a supreme Spirit.

So let's look at verse 2 and see what it tells us.

The earth was without form, and void; and darkness was on the face of the deep. And the Spirit of God was hovering over the face of the waters.

The earth was without form and void. this indicates that the earth was not solid as we know it today.

and darkness was on the face of the Earth this tells us that there was no light being reflected from it's surface.

And the Spirit of God was hovering over the face of the waters. This tells us that God was there observing his creation as it was in progress.

You might find this video interesting since it shows how verses 1 and 2 probably transpired.

http://hubblerevealscreation.com/genesis/

This brings us to a new question How could Moses have imagined the birth of the Earth and most probably the entire Universe?

My personal belief is that he was shown it by an eyewitness (God).

Hope this helps.

John 4:24 - God is spirit, and his worshipers must worship in the Spirit and in truth. The idea of a spirit by definition implies that there is not a corporeal body; one of the reasons why, when the Israelites created an idol to represent God, they were committing a form of blasphemy. When presented in corporeal form, our worship tends to go to that of an idol, rather than that of God.

In scripture, most of the language used to refer to "the right hand of God", etc. is metaphorical, which leads to the conclusion, that no, God does not have a physical body.

The most popular scripture to show that God does not have a body is John 4:24, which says

God is a Spirit: and they that worship him must worship him in spirit and in truth.

This does not exclude that he might also have a physical body. In Doctrine and Covenants 93:33 it states

For man is spirit. The elements are eternal, and spirit and element, inseparably connected, receive a fulness of joy;

from which it becomes clear that - from an LDS point of view - being a spirit does not conflict that one might have a physical body also. When it talks about the Spirit of God in the scriptures, then this means the Holy Ghost and not the spirit of God the Father (at least according to teachings of the Church of Jesus Christ of Latter-Day Saints).