Mark 6:13 - the apostles anointed the sick with oil and cured them. This is a sacrament of the Catholic Church instituted by Christ which heals us physically and spiritually.
James 5:14 - the presbyters (priests) are called to anoint the sick with oil and pray over them. Their sins are forgiven. This is the sacrament of the sick, also called extreme unction.
James 5:15 - during the sacrament of the sick, the priest's prayer of faith will "save" the sick man, and the Lord will raise him up. The word "save" comes from the Greek word "sozein" which means an eschatological saving of life from death.
James 5:14 -15 - these verses demonstrate another example of how priests effect the forgiveness of sins (here, even without confession) by the power of Jesus Christ. Protestants have no plausible exegesis of this passage other than to acknowledge the sacrament of the sick.
Gal. 4:13-14; 2 Tim. 4:20 - Paul was afflicted with sickness. These verses show that not all illnesses were cured in the apostolic age.
Tradition / Church Fathers
"For they are bold enough to teach, to dispute, to enact exorcisms, to undertake cures--it may be even to baptize." Tertullian, Prescription, 49 (A.D. 200).
"O God who sanctifiest this oil as Thou dost grant unto all who are anointed and receive of it the hallowing wherewith Thou didst anoint kings and priests and prophets, so grant that it may give strength to all that taste of it and health to all that use it." Hippolytus of Rome, Apostolic Tradition, 5:2 (c. A.D. 215) .
"In addition to these there is also a seventh [sacrament], albeit hard and laborious...In this way there is fufilled that too, which the Apostle James says: 'If then, there is anyone sick, let him call the presbyters of the Church, and let them impose hands upon him, anointing him with oil in the name of the Lord; and the prayer of faith will save the sick man, and if he be in sins, they shall be forgiven him.'" Origen, Homily on Leviticus, 2:4 (A.D. 244) .
"[The sick] considered a more terrible calamity than disease itself ... [instead of allowing] the hands of the Arians to be laid on the heads." Athanasius, Encyclical Epistle (A.D. 341) .
"[O]f the sacrament of life, by which Christians [baptism], priests [in ordination], kings and prophets are made perfect; it illuminates darkness [in confirmation], anoints the sick, and by its secret sacrament restores penitents." Aphraates the Persian Sage, Treatises, 23:3 (A.D. 345) .
"[this oil]...for good grace and remission of sins, for a medicine of life and salvation, for health and soundness of soul, body, spirit, for perfect strengthening." Serapion of Thmuis, Anaphora, 29:1 (A.D. 350) .
"For not only at the time of regeneration, but afterwards also, they have authority to forgive sins. 'Is any sick among you?' it is said, 'let him call for the elders of the Church and let them pray over him, anointing him with oil in the name of the Lord. And the prayer of faith shall save the sick, and the Lord will raise him up: and if he have committed sins they shall be forgiven him.'" John Chrysostom, On the Priesthood, 3:6 (A.D. 386) .
"They pray over thee; one blows on thee, another seals thee." Ephraim, Homily 46 (ante A.D. 373) .
"Why, then, do you lay on hands, and believe it to be the effect of the blessing, if perchance some sick person recovers? Why do you assume that any can be cleansed by you from the pollution of the devil? Why do you baptize if sins cannot be remitted by man? If baptism is certainly the remission of all sins, what difference does it make whether priests claim that this power is given to them in penance or at the font? In each the mystery is one." Ambrose, Penance, 1,8:36 (A.D. 390) .
"[I]f some part of your body is suffering...recall also the saying in the divinely inspired Scripture: 'Is anyone among you ill? Let him call the presbyters of the Church and let them pray over him, anointing him with oil in the name of the Lord. And the prayer of faith will save the sick man, and the Lord will raise him up, and if he be in sins they shall be forgiven (James 5:14-15)." Cyril of Alexandria, Worship and Adoration, 6 (A.D. 412) .
"[I]n the epistle of the blessed Apostle James...'If anyone among you is sick, let him call the priests...'. There is no doubt that this anointing ought to be interpreted or understood of the sick faithful, who can be anointed with the holy oil of chrism...it is a kind of sacrament." Pope Innocent [regn. A.D. 401-416], To Decentius, 25,8,11 (A.D. 416) .
"[L]et him who is ill receive the Body and Blood of Christ; let him humbly and in faith ask the presbyters for blessed oit, to anoint his body, so that what was written may be fufilled in him: 'Is anyone among you sick? Let him bring in the presbyters, and let them pray over him, anointing him with oil; and the prayer of faith will save the sick man, and the Lord will raise him up; and if he be in sins, they will be forgiven him (James 5:14-15)." Ceasar of Arles, Sermons, 13 (265), 3 (ante A.D. 542) .
"[A] priest is to be called in, who by the prayer of faith and the unction of the holy oil which he imparts will save him who is afflicted [by a serious injury or by sickness]." Cassiodorus, Complexiones (A.D. 570).