"The Council of Trent summarizes the Catholic faith by declaring: "Because Christ our Redeemer said that it was truly his body that he was offering under the species of bread, it has always been the conviction of the Church of God, and this holy Council now declares again, that by the consecration of the bread and wine there takes place a change of the whole substance of the bread into the substance of the body of Christ our Lord and of the whole substance of the wine into the substance of his blood. This change the holy Catholic Church has fittingly and properly called transubstantiation."" - Catechism of the Catholic Church 1376
In the Catholic Church, it is an official dogma that upon consecration by the priest, the bread and wine literally become the real, historical body of the Lord Jesus Christ. There are a large number of reasons to reject this claim, many of them Biblical, but there are also extra-Biblical reasons.
Biblically, this claim can be rejected simply because it is never actually found. The Council of Trent, and by extension the Catechism of the Catholic Church, may claim Jesus said it was truly His body that He was offering under the species of the bread, but what we actually see is "For I received from the Lord that which I also delivered to you: that the Lord Jesus on the same night in which He was betrayed took bread; and when He had given thanks, He broke it and said, “Take, eat; this is My body which is broken for you; do this in remembrance of Me.” In the same manner He also took the cup after supper, saying, “This cup is the new covenant in My blood. This do, as often as you drink it, in remembrance of Me.”" (1 Corinthians 11:23-25).
So what are we seeing here? Rather than Christ saying He is truly offering His body under the species of the bread, a phrase literally never attributed to Christ by any reliable source, we see that this is a ritual of remembrance.
Perhaps this is why, contrary to Catholic assertion, it has not always been the conviction of the Church of God that by the consecration of the bread and wine, the whole substance becomes His flesh and blood. First, this view was rejected by a wide range of Church "Fathers", including many that retain the title of "Saint" in the modern Catholic Church. Tertullian of Carthage wrote "They thought His discourse was harsh and intolerable, supposing that He had really and literally enjoined on them to eat his flesh, He, with the view of ordering the state of salvation as a spiritual thing, set out with the principle, It is the spirit that quickens; and then added, The flesh profits nothing — meaning, of course, to the giving of life." (1). Origen of Alexandria wrote "We have a symbol of gratitude to God in the bread which we call the Eucharist." (2). Augustine of Hippo (that's Saint Augustine) wrote "Understand spiritually what I have said; ye are not to eat this body which ye see; nor to drink that blood which they who will crucify Me shall pour forth." (3).
It was not until the Great Eucharistic Controversy in the 800s that Transubstantiation really became a thing. In 831 A.D.,a Catholic monk by the name of Paschasius Radbertus published his book "De corpore et sanguine Christi" (Concerning Christ's Body and Blood), in which he argued that the bread and the wine, upon consecration by a priest, are miraculously transformed into the actual body and blood of Jesus Christ. But a fellow monk by the name of Ratranmus disagreed, writing in his own book "The bread and wine are the body and blood of Christ in a figurative sense". In the modern day, it's hard to imagine two monks having such a different view, but during Ratranmus' life, Fourth Lateran Council (1215 A.D.) had not yet taken place, and so Transubstantiation was not yet an official Catholic dogma. That's not to say no one ever held to a similar view, but until more than 1,100 years after the death of John the Apostle, even Catholics were free not to.
In fact, according to the Council of Chalcedon (451 A.D.), they were right to reject it. Convening to address various heresies on the Trinity, and in particular Christology, the Council unanimously affirmed that Christ is "...to be acknowledged in two natures, inconfusedly, unchangeably, indivisibly, inseparably; the distinction of natures being by no means taken away by the union, but rather of the property of each nature being preserved, and concurring in one Person and one Subsistence, not parted or divided into two persons, but one and the same Son, and only begotten, God, the Word the Lord Jesus Christ; as the prophets from the beginning (have declared) concerning him, and the Lord Jesus Christ himself has taught us, and the Creed of the holy Fathers has handed down to us."
Although not directly addressing the issue of Transubstantiation, this "infallible" council has strong implications for it, as it affirms the Biblical truth that Christ does indeed have two natures (He is fully God and fully man), but those two natures are inconfusable, unchaneable, indivisible, and inseparable. In other words, Jesus' divine nature is not limited by His human nature, nor does His human nature take on the attributes of His divine nature.
With that in mind, we can refute Transubstantiation with a single scientific experiment: put it in your mouth! Every Sunday, millions of Catholics around the world gather to imagine they are eating the real, historical body of Jesus Christ, yet the vast majority of them will admit this isn't how it appears. The bread looks, sounds, smells, tastes, feels, and acts like bread. The wine, likewise, looks, sounds, smells, tastes, feels, and acts like wine. This is to the extent where those with certain medical conditions preventing them from eating bread, such as celiac disease, cannot safely consume it.
Yet, in order for the bread and wine to literally become Christ's flesh and blood, they would have to literally become Christ's flesh and blood. I can think of no other way to say it than that. See, if the Council of Chalcedon is correct, and Christ is truly to be acknowledged in two natures, inconfusedly, unchangeably, indivisibly, and inseparably, such that the properties of both natures are preserved, then He cannot possibly be bread, nor can bread be Him, for the same reason we cannot possibly be bread, nor can bread be us. Any attempt to say that the bread only appears to remain bread, but in reality is Christ, is to say that the property of Christ's human nature is no longer preserved.
But obviously, as Catholics often remind us, God is quite capable of performing some fantastic feats. Nothing is impossible for Him, after all. To that, first of all, I say wrong. Not wrong in the sense that I deny the phrase "nothing is impossible for God". This, after all, appears in Scripture. However, Scripture does also say "...it is impossible for God to lie..." (Hebrews 6:18), "...God, who cannot lie..." (Titus 1:2), and "...He cannot deny Himself." (2 Timothy 2:13). Thus, we must understand God's omnipotence in terms of sovereignty over His creation. He is in full control over what He creates, but He is not capable of changing His own nature. Effectively, God cannot cease to be God.
Now, with that in mind, I don't deny that God is perfectly capable of performing some fantastic miracles. I don't even deny that God could, if He desired, turn a human body into bread, or bread into a human body. However, first of all, we need some confirmation that He has done so. In the past, God has performed amazing miracles. He gave sight to a blind man, He walked on water, He calmed a storm, He even raised the dead. A common theme? When He did these things, there was no denying it.
Imagine, Jesus is sleeping on the boat, and the disciples wake Him saying "teacher, do you not care that we're going to die?" So Jesus walks up to the edge of the boat, tells the wind and the waves "be quiet", and immediately, the boat is capsized by a giant wave. Using a few of their last breaths before they drown, the disciples shout over another clash of thunder "who is this man that the wind and waves obey Him?"
The miracles God works simply do not work like that. They do not require faith for those present to believe. Rather, they are the proof that should lead to faith. Not that they always do, of course. Hard hearted men are always capable of saying that, rather than by the power of God, Jesus worked miracles in the name of Satan. But they were never capable of simply denying the miracle. They didn't look at the 3 day old corpse of Lazarus and find him still motionless, and very stinky. But we do look at the bread and wine and find them identical to their pre-consecration selves.
In fact, if we were to gather up many baskets of the Eucharistic host, some prior to consecration and some after, Catholics would never know the difference. You could ask them to point to the historical body of Christ, and their answers would vary according to the laws of probability. Occasionally, they'd guess correctly. Other times, not so much.
I dare to say, unless you count "every day" miracles, such as God upholding creation by the word of His power, there are no miracles that require faith to believe. But a Catholic may argue that we need faith for this one in particular, and it is quite acceptable for God to ask it of us. But going back to the Council of Chalcedon, not only should we expect some indication that God has indeed changed the host into the real flesh and blood of Christ, but He would be breaking His divine decree.
See, as much as I don't see the Council of Chalcedon as infallible, Catholics do, and it must be duly acknowledged that Biblically speaking, the logic is sound. Thus, if God cannot deny Himself, He cannot suddenly make Christ's human nature not human. This simply runs counter to who Christ was, and is, and is to be.
With all of this in mind, it is far better to take the Biblical approach to the Eucharist. We are to do it in remembrance of Christ, proclaiming His death until He comes. In so doing, we honor God, thanking Him for the sacrifice He made once for all, never needing to "re-present" it. As Christ says in John 6:35, "I am the bread of life. He who comes to Me shall never hunger, and he who believes in Me shall never thirst."
References
1. Tertullian of Carthage - On the Resurrection of the Flesh, Chapter 37
2. Origen of Alexandria - Against Celsus, book 8, chapter 57
3. Augustine of Hippo - Expositions on the Psalms