Below we will look at the teaching of Christ, the teaching of Scripture, and the teaching of Church Fathers.
John 3:5: Jesus answered, Verily, verily, I say unto thee, Except a man be born of water and of the Spirit, he cannot enter into the kingdom of God.
Matthew 12:30: He that is not with me is against me; and he that gathereth not with me scattereth abroad.
Matthew 7:14: Because strait is the gate, and narrow is the way, which leadeth unto life, and few there be that find it.
John 3:36: He that believeth on the Son hath everlasting life: and he that believeth not the Son shall not see life; but the wrath of God abideth on him.
John 6:47: Verily, verily, I say unto you, He that believeth on me hath everlasting life.
John 8:24: I said therefore unto you, that ye shall die in your sins: for if ye believe not that I am he, ye shall die in your sins.
John 14:6: Jesus saith unto him, I am the way, the truth, and the life: no man cometh unto the Father, but by me.
Acts 4:12: Neither is there salvation in any other: for there is none other name under heaven given among men, whereby we must be saved.
Acts 16:31: And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
According to Romans 10:9, one must confess "Jesus is Lord" and believe that God has raised him from the dead.
In Galatians 5:19-20, St. Paul says heretics will not inherit the Kingdom of God. If heresy prevents people from salvation, then what does that say about people in religions that don't acknowledge Christ?
1 John 5:12: He who has the Son has life; he who does not have the Son of God does not have life.
1 John 2:22: Who is a liar but he that denieth that Jesus is the Christ? He is antichrist, that denieth the Father and the Son.
1 John 2:23: Whosoever denieth the Son, the same hath not the Father: (but) he that acknowledgeth the Son hath the Father also.
1 Timothy 2:5: For there is one God, and one mediator between God and men, the man Christ Jesus;
Hebrews 8-10 teaches that Christ is the only Mediator between God and humanity.
Ephesians 4:5: One Lord, one faith, one baptism,
Reconciliation with God is only for those "in Christ." (2Cor. 5:17). The phrase "in Christ" is never used in Scripture of anyone but believers.
Jesus taught that they that worship God must worship him in spirit and in truth. (John 4:24). Not only do other religions not worship God in truth, but they contradict each other.
Athanasian Creed: "Whosoever will be saved, before all things it is necessary that he hold the catholic faith. Which faith except every one do keep whole and undefiled; without doubt he shall perish everlastingly."
The Third Council of Constantinople (680): In no other than the orthodox faith could men be saved.
The Second Council Of Constantinople, Sess. 3, AD 553: “… the holy, catholic and apostolic church of God, if anyone separates himself from its communion by holding contrary opinions, such a person, since he alienates himself from the orthodox faith and numbers himself with the heretics, is justly condemned and anathematized by the holy Church of God.” Notice, people are separated from the Church by holding contrary opinions.
St. John Chrysostom: "We know that salvation itself is a property of the One Church, and that no one can be outside of the catholic Church and yet share the Faith of Christ, or be saved...Neither do we offer any part of that hope to the ungodly heretics, but we place them entirely outside of that hope; indeed, they have not the least participation in Christ, but vainly assume for themselves that saving Name.” (Migne P. G. 59:725).
St. John Chrysostom: “The universal [catholic] Church is a great paradise...and should anyone be found in the Church ailing with heretical error from the teaching of the serpent...then he is cast out of this paradise, even as Adam was cast out from the paradise [of old].” (PG 59:545CD).
St. Theodore the Studite: “Chrysostomos loudly declares not only heretics, but also those who have communion with them, to be enemies of God.” (Epistle to Abbot Theophilus).
St. Theodore the Studite: “Guard yourselves from soul-destroying heresy, communion with which is alienation from Christ.” (P.G. 99.1216).
Blessed Augustine: "Outside the Catholic Church everything may be had except salvation. One may have orders and Sacraments, one may sing Alleluia and answer Amen, one may hold the Gospel, one may have and preach in the name of the Father, the Son, and the Holy Ghost, but nowhere except in the Catholic Church can one find salvation." (PL. Vo. 43, pp. 689-698).
St. Fulgence of Ruspe (468-533 A.D.): "Hold most firmly and never doubt in the least that no person baptized outside the Catholic Church can become a participant of eternal life if, before the end of this life, he has not returned and been incorporated into the Catholic Church." (Migne, PL Vo. 65. pp. 671-706).
St. Pope Gregory the Great (540-604 A.D.): "The holy universal Church teaches that it is not possible to worship God truly except in her and asserts that all who are outside her will not be saved." (Moralia, Lib. XIV, Cap. V. n. 5. PL, Tom. LXXV. col. 1043).
St. Lactantius (240-320 A.D.): "Therefore, it is the Catholic Church alone which retains true worship. This is the fountain of truth, this is the abode of the faith, this is the temple of God; into which if any one shall not enter, or from which if any shall go out, he is estranged from the hope of life and eternal salvation." (Divinarum Institutionum--The Divine Institutes, Book IV, Chapter 30: See also: Ante-Nicene Fathers, Lactantius, Vol. 7. Edited by Philip Schaff, Hendrickson Publishers, Peabody. Mass., Second Printing, 1999, p. 133).
St. Jerome: “The priest made you see your sins and your leprosy, and led you back into God’s assembly through blood and water: through blood, the Passion of Christ; through water, by baptism. After you have contracted the leprosy of corruption, you cannot be healed save through the Blood of Christ and through baptism… as long as you do not come to blood and water, you cannot be saved.” (Homily On Psalm 84).
St. Maximus of Turin, c. AD 423: “For before a catechumen is baptized, he lies like water, cold and pallid, but a member of the faithful is vigorous and ruddy like wine. A catechumen, I say, is like water that has no taste, no smell, and no value; which avails not for use, delights not for refreshment, and passes not to be kept. For just as water, when kept too long, turns interiorly worse, rots and stinks, so too the catechumen, when he remains a catechumen for too long, turns interiorly worse, becomes worthless and is lost. For the Lord said, ‘Unless one is born again of water and the Holy Spirit, he will not enter the Kingdom of Heaven.’ (Jn 3:5) But he who does not enter the Kingdom necessarily remains in Hell.” (Sermon 65).
The following quotes are from the book, "A Dictionary of Early Christian Beliefs," by Bercot. The sources are provided in the book.
St. Lactantius: ..."This is the everlasting temple. If anyone has not sacrificed in this, he will not have the reward of immortality"...
St. Cyprian: "Likewise, neither can he be saved by baptism who has not been baptized in the church."
St. Cyprian: "There is no salvation outside of the church."
St. Cyprian: ..."remission of sins is not granted except in the church."
St. Cyprian: "The house of God is one, and there can be no salvation to anyone except in the church."
(End of quotes from “A Dictionary of Early Christian Beliefs”)
St. Cyril of Jerusalem: ‘There is one Lord, one Faith, one baptism’ [Eph. 4:5], and this is what we believe with all our soul, with all our heart, and with all our mind, that there is salvation in none other than in Jesus Christ the Nazerene [cf. Acts 4:10-12]. And this is what we piously believe and follow. And we recognize that works without true doctrine are not accepted by God, neither is true doctrine without works accepted by God. For what profit is it, to know well the doctrines concerning God, and yet to be a vile fornicator? And again, what profit is it, to be nobly temperate, and an impious blasphemer? A most precious possession therefore is the knowledge of doctrines: also there is need of a wakeful soul, since there are many that make spoil through philosophy and vain deceit. The Greeks on the one hand draw men away by their smooth tongue, for honey droppeth from a harlot’s lips: whereas they of the Circumcision deceive those who come to them by means of the Divine Scriptures, which they miserably misinterpret though studying them from childhood to all age, and growing old in ignorance. But the children of heretics, by their good words and smooth tongue, deceive the hearts of the innocent, disguising with the name of Christ as it were with honey the poisoned arrows of their impious doctrines: concerning all of whom together the Lord saith, Take heed lest any man mislead you. This is the reason for the teaching of the Creed and for expositions upon it.” (St. Cyril, Archbishop of Jerusalem, Catechetical Lectures - Lecture 4:
On Ten Points Of Doctrine).
St. Cyril of Jerusalem: "The method of godliness consists of these two things, pious doctrines and virtuous practice." (Catechetical Lectures 4.2). Jaroslav Pelikan: "neither of which was acceptable to God without the other". (Pelikan, The Christian Tradition: A History of the Development of Doctrine, Vol. 1: The Emergence of the Catholic Tradition p. 2).
In The Spiritual Meadow of St. John Moschus, we read: “Once a monk called Theophan came to see the great elder Kyriakos...” (He tells the elder that in his country he is in contact with Nestorians whereupon) “the elder begins to try to convince the monk of his error and to pray that he abandon that fatal heresy and join himself to the holy catholic and apostolic Church.” “It is impossible to be saved (‘without right belief).’” (The monk is interested and the elder offers him his cell saying:) “I have hope that God in His mercy will reveal the truth to you.’” “And leaving the monk in his cave, the elder set out for the Dead Sea, praying for the monk as he went. And indeed the next day about the ninth hour the monk sees someone, strange in appearance, who says to him, ‘Come and find out the truth.’ And taking him he leads him to a gloomy, stinking place emitting flames and shows him Nestorius and Theodore (of Mopsuestia), Eutyches and Apollonarius, Evagrius and Didymus, Dioscorus and Severus, Arius and Origen, and others. And pointing at them he says to the monk, ‘That is the place prepared for heretics and those who taught falsely about the Mother of God and those who follow their teachings. If you do not want to taste the same punishment turn to the holy catholic and apostolic Church to which the elder who is instructing you belongs. I tell you: even though a man be adorned with all the works of charity, but does not have right belief he will find himself in that place.’ “With these words the monk came to himself. When the elder returned the monk told him everything that he had seen and in a short time joined himself to the holy, catholic, and apostolic Church. Staying in the monastery of Kalamon he lived with the elder for some years and died in peace.” (St. John Moscus, Spiritual Meadow; cited in “Commentary on the Latest Recommendations of the ‘Joint Commision For Theological Dialogue Between the Orthodox And Oriental Churches," Orthodox Life, vol. 42, no. 3, May-June 1991, pp. 5-18.; quotation appears on p. 17).