The earliest citations of 2 Timothy according to biblindex. This took me a while to compile especially given the fact that biblindex doesn't include Irenaeus Against Heresies in their compilation (for reasons I don't understand).
2 Timothy 1, 2 Origen
2 Timothy 1, 3 Origen
2 Timothy 1, 6 Origen
2 Timothy 1, 7,8 Clement of Alexandria (Stromata 4.49.5) ["Be not therefore ashamed of the testimony of our Lord, or of me his prisoner," he writes to Timothy] BP1, Origen, Tertullian (Scorpiace)
2 Timothy 1, 10 Irenaeus (Demonstration), Clement of Alexandria (Stromata 4.91.2), Origen
2 Timothy 1, 11 Acts of Paul (p.50, l.15) BP1, Origen, Novatian
2 Timothy 1, 13 Origen
2 Timothy 1, 14 Origen, Tertullian (Prescription 25.2 p.206, l.7) [and again: "That good thing which was committed unto thee keep."]
2 Timothy 1, 15 Tertullian (Aduersus Hermogenem 1.2, Resurrection of the Flesh 24.8) not in biblindex Prescription " ... that certain men, like Phygellus, and Hermogenes, and Philetus, and Hymenµus, deserted His apostle."
2 Timothy 1, 16 Acts of Paul B 2 .2 (p.148, l.5) Tertullian Resurrection of the Flesh 23.10) Ignatius (Epistle to the Smyrneans 10.2) not in biblindex Ignatius Epistle to the Ephesians ["hath not been ashamed of my chain,"] Origen
2 Timothy 1, 18 Origen not in biblindex Ignatius to the Smyrnaeans ["The Lord grant" to you "that ye may find mercy of the Lord in that day!"] Ignatius Epistle to Hero [Salute Cassian, my host, and his most serious-minded partner in life, and their very dear children, to whom may "God grant that they find mercy of the Lord in that day,"]
2 Timothy 2, 1 Minucius Felix 37.3, Clement of Alexandria (Stromata 1.3.3) [And the things which thou hast heard of me among many witnesses, the same commit thou to faithful men, who shall be able to teach others also."] Hippolytus Antichrist
2 Timothy 2, 2 Clement Hypotyposeis (p.201, l.13) BP1, Clement of Alexandria (Stromata 1.3.3) Hippolytus Antichrist, Tertullian (Prescription 25.6,8) [publicly: "Before many witnesses" is his phrase.] [" Nor, again, must the circumstance of his having wished him to "commit these things to faithful men, who should be able to teach others also,"] Origen
2 Timothy 2, 3 Acts of Paul 10.2, Tertullian De exhortatione castitatis 12.1, Origen, not in biblindex Tertullian Prescription [against those who said that "the resurrection was past already."]
2 Timothy 2, 4 Minucius Felix 37.3, Acts of Marcellus, Origen, Tertullian De exhortatione castitatis 12.1, Cyprian, Ignatius to Polycarp 6.2, not in biblindex Ignatius to Hero ["No man that warreth entangleth himself with the affairs of this life, that he may please him who hath chosen him to be a soldier; and if a man also strive for masteries, yet is he not crowned except he strive lawfully."]
2 Timothy 2, 5 Pseudo-Clement of Rome Ad uirgines epistulae duae 1.5, Hippolytus, Origen, Cyprian
2 Timothy 2, 7 Pseudo-Clement of Rome Ad uirgines epistulae duae 1.3.5
2 Timothy 2, 8 Acts of Paul of Samosata, De Recta in Deum Fide, Anti-Paul of Samosata to the Antiochenes.
2 Timothy 2, 11 Origen, Tertullian Scorpiace 13.11
2 Timothy 2, 12 Polycarp to the Philippians 5.2 [worthily of Him, "we shall also reign together with Him,"], Cyprian, Hippolytus, Origen, Hagiographic writing.
2 Timothy 2, 13 Origen, Tertullian Scorpiace 13.11
2 Timothy 2, 14 Origen
2 Timothy 2, 15 Acts of Paul 10.1, Clement of Alexandria Stromata 1.3.3 [" Accordingly, the blessed apostle very appropriately and urgently exhorts us "not to strive about words to no profit, but to the subverting of the hearers, but to shun profane and vain babblings, for they increase unto more ungodliness, and their word will eat as doth a canker."], Pseudo-Clement of Rome Ad uirgines epistulae duae 1.3.5, Origen
2 Timothy 2, 16 Clement Stromata 1.49.3
2 Timothy 2, 17 Clement Stromata 1.49.3, Tertullian Prescription 7.7, Hippolytus, Cyprian, Origen not in biblindex Irenaeus Against Heresies [rom being able to raise the dead, as the Lord raised them, and the apostles did by means of prayer, and as has been frequently done in the brotherhood on account of some necessity-the entire Church in that particular locality entreating [the boon] with much fasting and prayer, the spirit of the dead man has returned, and he has been bestowed in answer to the prayers of the saints-that they do not even believe this can be possibly be done, [and hold] that the resurrection from the dead] NOT A CITATION PER SE
2 Timothy 2, 18 Acts of Paul 2.14, Tertullian 33.7, Gospel of Philip 90, Hippolytus
2 Timothy 2, 19 Egerton papyrus, Origen
2 Timothy 2, 20 Refutation of All Heresies 5.7.36, Gospel of Philip 51, Origen, Cyprian
2 Timothy 2, 21 Acts of Thomas, Gospel of Philip 51, Origen
2 Timothy 2, 22 Clement of Alexandria Stromata 1.51.2, Origen
2 Timothy 2, 23 Pseudo-Clement of Rome Ad uirgines epistulae duae 1.10.4, Clement of Alexandria Stromata 1.51.2, 5.5.1, Origen, Cyprian, not in biblindex Irenaeus Against Heresies [of truth, hold no such opinions, but that they did also preach to us to shun these doctrines] NOT A CITATION PER SE
2 Timothy 2, 24 Origen, Cyprian not in biblindex Ignatius Ephesians [Wherefore Paul exhorts as follows: "The servant of the Lord must not strive, but be gentle towards all men, apt to teach, patient, in meekness instructing those that oppose themselves."]
2 Timothy 2, 25 Polycarp to the Philippians 11.4, Origen
2 Timothy 2, 26 Origen not in biblindex Ignatius Philadelphians [I therefore exhort you in the Lord to receive with all tenderness those that repent and return to the unity of the Church, that through your kindness and forbearance they may recover[14]]
2 Timothy 3, 1 Tertullian De ieiunio (aduersus psychicos) 12.2, On the Resurrection of the Flesh 41.6, Hippolytus, Origen
2 Timothy 3, 2 Clement of Alexandria Stromata 1.87.7, Origen
2 Timothy 3, 4 Origen, Gregory in Praise of Origen 12.149 not in biblindex Ignatius Magnesians [who are "lovers of pleasure, and not lovers of God, having a form of godliness, but denying the power thereof."]
2 Timothy 3, 5 Acts of Paul, Pseudo-Clement of Rome Ad uirgines epistulae duae 1.3.3 - 4, 1.10.1, 1.10.5, Cyprian, Origen
2 Timothy 3, 6 Origen not in biblindex Irenaeus Against Heresies 1 [Such are the words and deeds by which, in our own district of the Rhone, they have deluded many women, who have their consciences seared as with a hot iron.]
2 Timothy 3, 7 Hippolytus, Origen not in biblindex Irenaeus Against Heresies 4 [For, being driven away from Him who truly is [God], and being turned backwards, he shall be for ever seeking, yet shall never find out God;] 5 [Now, such are all the heretics, and those who imagine that they have hit upon something more beyond the truth, so that by following those things already mentioned, proceeding on their way variously, in harmoniously, and foolishly, not keeping always to the same opinions with regard to the same things, as blind men are led by the blind, they shall deservedly fall into the ditch of ignorance lying in their path, ever seeking and never finding out the truth]
2 Timothy 3, 8 Origen
2 Timothy 3, 9 Acts of Paul, Origen
2 Timothy 3, 11 Acts of Paul, Origen
2 Timothy 3, 12 Gaius of Rome fragments, Hippolytus Against Gaius, Ad uirgines epistulae duae 2.1.1, Origen
2 Timothy 3, 13 Gaius of Rome fragments, Hippolytus Against Gaius, Origen, Gregory Thaumaturgus 13.156
2 Timothy 3, 14, 15 Clement of Alexandria Exhortation 87.1
2 Timothy 3, 16 Clement of Alexandria Exhortation 87.2, Stromata 7.101.5, To Theodore, Tertullian De cultu feminarum 1.3.3, Origen, De Recta in Deum Fide
2 Timothy 3, 17 Clement of Alexandria Exhortation 87.2, Pseudo-Clement of Rome Ad uirgines epistulae duae 1.2.5, 1.9.2,
2 Timothy 4, 1 Didascalia apostolorum 19, 26, Hippolytus, Dionysius of Alexandria
2 Timothy 4, 2 Origen
2 Timothy 4, 3 Tertullian Prescription 7.1, Origen, Cyprian not in biblindex Irenaeus Against Heresies 2 [according to the meaning of the Greek word, because she secretly stirred up men), without the knowledge of the Demiurge, to give forth profound and unspeakable mysteries to itching ears.]
2 Timothy 4, 4 Origen, Cyprian
2 Timothy 4, 5 Origen,
2 Timothy 4, 6 Tertullian Scorpiace 13.10, Origen, Ignatius to the Romans 2.2 not in biblindex Ignatius Antiochenes [till God shall show who is to hold the rule over you. For "I am now ready to be offered,"]
2 Timothy 4, 7 Testimony of Truth 34.10, Acts of Thomas, Pseudo Clement of Rome Ad uirgines epistulae duae 1.6.4, 2.15. 6, Origen, Hippolytus, Life of Cyprian
2 Timothy 4, 8 Sybilline Oracle 2, Clement of Alexandria Instructor 2.74.1, Cyprian, Origen, Acts of Thomas, Acts of Thomas, Pseudo Clement of Rome Ad uirgines epistulae duae 2.15.6, Tertullian Scorpiace 13.10, Tertullian De pudicitia 10.8, Hippolytus, Life of Cyprian
2 Timothy 4, 10 Acts of Paul, Polycarp to the Philippians 9. 2, Origen not in biblindex Irenaeus Against Heresies 3 [and is departed unto Thessalonica; Crescens to Galatia, Titus to Dalmatia. Only Luke is with me."]
2 Timothy 4, 11 Acts of Paul, Origen
2 Timothy 4, 13 Tertullian De Oratione 15.2
2 Timothy 4, 17 Hippolytus
2 Timothy 4, 19 Acts of Paul
2 Timothy 4, 22 Acts of Paul
I for one am surprised by the combination (1) reference in Polycarp but (2) lack of references in Irenaeus. That might be important. The only direct citation I see is from the end of the epistle - "only Luke is with me" and the reference to Paul knowing Linus - which is of course very significant for the introduction of the gospel of Luke.
I will examine Polycarp's use of 1 Timothy later but for the moment it can't be overstated how important the radical new understanding of Paul is in 1 Timothy and its 'influence' on Irenaeus. I would rather suppose that Irenaeus wrote 1 Timothy to make Paul a mirror of his own inclinations. I suspect that Polycarp wrote 2 Timothy but let's wait and see ...