While this may be LDS doctrine, the experience of Orthodox Christians is one routinely reinforced by actual evidence of God’s existence and of the interaction of His uncreated grace with His creation.
This is because we believe in an incarnate God who created all things (John 1:1-14), including time, space and the universe, and who is omnipresent, omniscient, omnipotent, impassible, eternal and immutable, distinct from creation but not outside of it.
Additionally we believe the three persons of the all-Holy consubstantial and life-giving Trinity are of one essence with the Father, from whom the Son is begotten before all ages, and from whom the Holy Spirit proceeds, and are coeternal, coequal, and united in a state of perfect love, and the Son furthermore is fully man and fully God, his humanity and divinity hypostatically united without change, confusion, separation or division.
Thus, while God in His essence is beyond human comprehension, we can perceive and interact with His uncreated energies, including the grace of the Holy Spirit, and we furthermore have many sacred relics which exhibit miraculous properties of which the Ark would simply be another example. Indeed the Ethiopian Tewahedo Orthodox Church is in posession of the Ark of the Covenant. The relics of St. Nicholas of Myra are among several relics and icons which gush sacred myrhh. And the fragments of the True Cross are known for their ability to heal the sick; indeed the Cross was identified by being used to resurrect two recently deceased people when St. Helena conducted what was arguably the first archaeological survey in history, of Jerusalem, after her son Emperor Constantine converted to Christianity (which she had previously professed), enabling her to travel to the Hagiopolis and rebuild it, for it was still largely in ruins, having never been entirely rebuilt following the failed Bar Kochba revolt in 130 AD.
I don’t know if Mormons have any miraculous objects in their temples or not, but we do in ours (and additionally many of our relics were stolen by the Venetians and other Roman Catholic temples, or were in the possession of the Roman church before the Great Schism estranged them from us, so Orthodox pilgrims will visit Roman churches to venerate the many holy relics and icons in their possession; there are also a small number of holy relics, icons and important historical artifacts in Anglican, Lutheran and other Protestant churches).
Imagine a deist complaining of "extra-Biblical nonsense." 
