I will answer the question from the viewpoint of Roman Catholic doctrine.
It is de fide Catholic doctrine that the souls of those who die in original sin only go to an everlasting state , usually called "Limbo of Infants" (Latin limbus infantium or limbus puerorum), in which they do not see God and do not have any expectation of being redeemed and taken to the vision of God at some future time. It is sententia communis, not de fide, that the souls in such state do not suffer any torment.
However, it is not de fide doctrine that there are actually any souls in that state, because it is not de fide doctrine that any person actually dies in original sin only, see section 3.
2. Magisterial definitions about the fate of those who die in original sin only
The doctrine that those who die in original sin only go to an everlasting state of privation of the Beatific Vision, commonly called limbo of infants, is de fide, having been taught in the following magisterial definitions . Note that the most authoritative definitions, i.e. those of Ecumenical Councils, do not state explicitely that souls in the limbo of infants do not suffer any torment or fire, so that this specific point is not de fide but only sententia communis.
Pope Innocent III, letter "Maiores Ecclesiae causas" to Imbert, archbishop of Arles, 1201, DS 780 Dz 410:
The punishment of original sin is deprivation of the vision of God,
but the punishment of actual sin is the torments of everlasting hell.
Ecumenical Council of Lyon II, Profession of Faith of Michael Palaeologus, 1274, DS 858 Dz 464:
The souls of those who die in mortal sin or with original sin only,
however, immediately descend to hell, yet to be punished with
Ecumenical Council of Florence, Session 6 — 6 July 1439, promulgated as bull "Laetentur Caeli: Bulla Unionis Graecorum" by Pope Eugene IV, DS 1306 Dz 693:
Moreover, the souls of those who depart in actual mortal sin or in
original sin only, descend immediately into hell, yet to be punished
with different punishments.
Illorum autem animas, qui in actuali mortali peccato vel solo
originali decedunt, mox in infernum descendere, penis tamen disparibus
Pope Pius VI, Constitution "Auctorem fidei," 28 Aug. 1794, listing condemned propositions of the Synod of Pistoia, DS 2626 Dz 1526:
- The doctrine which rejects as a Pelagian fable, that place of the lower regions
(which the faithful generally designate by the name of
the limbo of children) in which the souls of those departing with the
sole guilt of original sin are punished with the punishment of the
condemned, exclusive of the punishment of fire, just as if, by this
very fact, (that) these who remove the punishment of fire introduced
that middle place and state free of guilt and of punishment between
the kingdom of God and eternal damnation, such as that about which the
Pelagians idly talk,--false, rash, injurious to Catholic schools.
Since the original of the text in italics is perinde ac si hoc ipso quod qui poenam ignis removent, the "that" in the usual English translation does not belong.
Note that the Pistoians did not hold that limbo, not featuring fire, was a "middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation", but on the contrary, accused those who held that limbo did not feature fire of holding that. That is, the Pistoians held that the limbo of infants did feature fire and rejected as a Pelagian fable the (wholly legitimate) position that it did not.
3. The question of whether the infants who die unbaptized die in original sin
It is evident that, in order to state positively that an infant who dies upbaptized dies in a state of original sin, it is necessary to state positively that God, in the last second of the infant's life, does not reveal Himself and his love directly to the soul of the dying infant and asks him or her for a response to that revelation, so that if the infant accepts God's love and turns to God, it is a case of baptism of desire whereby God infuses sanctifying grace and charity to the soul of the infant and he or she goes to the Beatific Vision.
The hypothesis that God performs that revelation in the last instant of a dying infant's life is called "illumination theory", and while it is obvious that it cannot be affirmed that this is the actual case, it is also obvious that it cannot be affirmed that this is not the actual case .
The possible objection that the last second of an infant's life is too short a time for the divine revelation and the infant's response can be addressed very easily. It is well known that, according to ordinary laws of physics, time flows at different "speeds" in different parts of the universe, e.g. those where the gravitational field has different strength (which is taken into account in the GPS system). From that, it is easy to see that God can change the time scale of the child's soul with respect to the rest of the universe, so that a whole day elapses for the soul of the dying child while only a second elapses in the rest of the universe.
Moreover, that was probably the case when the sun stood still for a day in Gibeon at the order of Joshua (Josh 10:12-14): God did not stop the Earth's rotation (or the sun's movement, for the geocentrics out there), but changed the scale of time flow in the battlefield 100,000 to 1 with respect to the rest of the universe. Why didn't God just kill all the Amorites in a flash? The message of the passage is loud and clear: if God wants the collaboration of his chosen ones in order that they win a battle, He will give them the time they need for that collaboration, even if it requires changing the scale of their time flow 100,000 to 1 with respect to the rest of the universe.
 Which, after the resurrection of the dead, becomes the state of the whole person, soul and body.
 DS = Denzinger-Schönmetzer; Dz = Denzinger.
[2.a] Denzinger-Schönmetzer 1963:
[2.b] Denzinger 1954:
 Brian W. Harrison, O.S., 2005, "Could Limbo Be 'Abolished'?".
The article, which does a good review of Catholic doctrine, says about the illumination theory:
"Are we to suppose that God miraculously 'fast-forwards' the mental development of these infants (and gravely retarded persons) in the instant before death, following this up with a special illumination so as to render them capable of an at least implicit desire for baptism? But miracles cannot be gratuitously postulated, so we could never be sure, in the absence of any revealed truth in Scripture or Tradition, that this is in fact what God does."
The obvious reply is: "Are we instead to suppose that God does not do that?" Because just as we cannot be sure that this is in fact what God does, we cannot either be sure that this is not in fact what God does.