Does the Bible ever say that the Ark of the Covenant flew or levitated?

In regards to the above text: "Consider Numbers 10:33-36, the ark seems to be flying in this text". I think that your statement is somewhat interpretive of the text. Nothing is hidden in the Bible. You can take the text at face value. If you do, then the above passages do not mention the ark flying by any means. When I read that passage it would appear to me that Israelites in charge of carrying the Ark scouted 3 days ahead of the main tribe and that Moses' blessed the Ark upon exit and return. I think its these type of passages that can be twisted to create alternate paradigms that seem plausible to luke warm Christians (of which I have been guilty of as well). That said, The "History Channel"or the Atheist Channel as I like to call it is no source of viable, accurate scripture for Christians. Shows like Ancient Aliens are clever at throwing in Bible passages and references (They had a great time fabricating stories when they discussed the book of ezekiel) to muddy the waters,but even mainstream scientists say the work is largely founded on wild fantasies that have no merit in the scientific world. Also regard these type of programs with respect as they are undoubtedly one of the largest tools that diseminate lies creating doubt, traps and pitfalls for Christians.

Jefferson

Matt 6:34

I think the OP might be referring to this episode on the History channel
https://youtu.be/2kW-yMvlAqQ?t=2177
Not sure where this text is from however, mentioned in the video:
"God said to him: Uzzah, it bears those that carry it, can it not certainly bear itself".

I have no idea what OP might have heard about this, but the somewhat surprising answer to the question:

Does the Bible ever say that the Ark of the Covenant flew or levitated?

is: Yes.

There is a curious verse embedded in the story of the miraculous crossing of the Jordan River by the Israelites in Joshua 3-4. The miracle of the stopped river is bounded by the movements of the priests bearing the ark, beginning in 3:13 -

It shall come about when the soles of the feet of the priests who carry the ark of the Lord, the Lord of all the earth, rest in the waters of the Jordan, the waters of the Jordan will be cut off, and the waters which are flowing down from above will stand in one heap. _[NASB]

The miraculous moment ends with the priests and the ark moving out of the stream-bed on to dry land in 4:18 -

It came about when the priests who carried the ark of the covenant of the Lord had come up from the middle of the Jordan, and the soles of the priests’ feet were lifted up to the dry ground, that the waters of the Jordan returned to their place, and went over all its banks as before. _[NASB]

I have chosen to cite the NASB because its rendering is quite literal. Many English versions do not render 4:18 this way, but often go with something like:

• HCSB: "...and their feet stepped out on solid ground,..."
• NIV: "...No sooner had they set their feet on the dry ground..."
• NRSV: "...and the soles of the priests’ feet touched dry ground,..."

for the Hebrew at that point is curious. It unambiguously presents the priests not as carrying the ark but as being carried:

...and the soles of the priests’ feet were lifted up to the dry ground,...
נִתְּק֗וּ כַּפּוֹת֙ רַגְלֵ֣י הַכֹּהֲנִ֔ים אֶ֖ל הֶחָרָבָ֑ה

The verb here, NTQ, is quite clearly Niphal, i.e., the priests' feet are "passive", as in NASB's rendering above, and cf. the BDB entry:

Now this is a bit embarrassing, really, or at least awkward. What could possibly lift up the priests' feet in this way?! The Septuagint translator certainly avoided it (or, as Holmes thinks, didn't understand it), as have most commentators, antique or modern (and I have checked Keil, Cooke, Boling, Soggin, Woudstra, Butler, Hess, Pitkänen, and Rösel -- not one comments on this!). And the only academic article I'm aware of on the wider theme neglects this completely: G. Coats, "The Ark of the Covenant in Joshua: A Probe into the History of a Tradition", Hebrew Annual Review 9 (1985): 137-157. But not Rashi, who had this to say about it:

the soles of the priests’ feet were lifted up: from the water to the dry land which was beside them, and the waters returned to their place. Hence, the Ark is found to be on one side and Israel on the other side. Accordingly, the Ark lifted its bearers and passed over. And concerning this matter, Uzzah was punished when he took hold of the Ark. If it bore its bearers, can it not be deduced by a fortiori conclusion that it can bear itself?

It is interesting that Rashi uses this incident -- that it was, in fact, the power of the Ark itself that carried the priests out of the river bed -- to explain the LORD's anger against Uzzah who steadied the ark in 2 Samuel 6 (see answer above). Of course, Uzzah need not have intevened in this way, since the Ark had previously demonstrated its capacity to "bear itself", as Rashi put it.

So this, then, is the surprising moment when the Ark of the Covenant "flew or levitated" in a story in the Hebrew Bible.

_{N.b. I'm resisting the temptation to add a graphic to this answer. Caveat lector.}

The Ark of the Covenant is never recorded as having flown in Scripture.

There were only two means by the Ark ever moved -

1. Per God's command, the Ark was transported by the priests, using the Acacia poles which held it through the base, for this purpose.

   When the Priests carried it into the Jordan River when it was still flowing in Joshua 5, for example, the river receded. If it could have flown, there would have been no need for the priests to "step out" in faith.

   

2. On two occasions, contrary to God's command, an ox cart was used to transport it.

   a. In 2 Samuel 6, the Ark is being drawn "on a new ox cart," as follows:

   And they carried the ark of God on a new cart and brought it out of the house of Abinadab, which was on the hill. And Uzzah and Ahio,[a] the sons of Abinadab, were driving the new cart,[b] 4 with the ark of God, and Ahio went before the ark.

   Famously, it began to tip, and Uzziah tried to steady it. For this, he was immediately zapped (that's not a technical term) by God, and died instantly. Had the Ark been capable of flight, one would have assumed that it would have steadied itself in such an instance.

   

   b. Also, in 1 Samuel 5, the Israelites had lost the Ark in battle to the Philistines. When the Philistine's local deity Dagon kept falling over before it, as if it were bowing, the Philistines started moving it from Ashdod to Gath and throughout their cities. Problem was- people kept dying. So, eventually they decided to get rid of it, and send it on its merry way with a bunch of gifts for the Israelites.
   

   1 Samuel 6 records:

   The men did so, and took two milk cows and yoked them to the cart and shut up their calves at home. 11 And they put the ark of the Lord on the cart and the box with the golden mice and the images of their tumors. 12 And the cows went straight in the direction of Beth-shemesh along one highway, lowing as they went. They turned neither to the right nor to the left, and the lords of the Philistines went after them as far as the border of Beth-shemesh. 13 Now the people of Beth-shemesh were reaping their wheat harvest in the valley. And when they lifted up their eyes and saw the ark, they rejoiced to see it. 14 The cart came into the field of Joshua of Beth-shemesh and stopped there. A great stone was there. And they split up the wood of the cart and offered the cows as a burnt offering to the Lord.

3. The story of the Ark of the Covenant being moved by Nazi jeep pickup truck is, alas, non-canonical, and adheres only to the obscure Spielbergian sects of Christianity.

   

In all seriousness, however, the legends of the ark flying may be attributable to the winged cherubim on top of the mercy seat. This ornament of two angels, only either end of the ark, their wings touching, may have inspired such an idea, but they were strictly ornamental.