Augustine's phrase Cum dilectione hominum et odio vitiorum is an early origin for what became "love the sinner, hate the sin," but the phrasing as we know it may have entered English in the 17th century. For this answer, I will trace the two expressions "love the sinner" and "hate the sin" (and particularly the apposition of sinner and sin) to early citations in English. In other words, what is the more literal origin in English?
The Earliest Possible
The phrasing would not have been possible in English before the 14th century, when sinner first enters the lexicon (Oxford English Dictionary). My searches of the Middle English Corpus for sin/ner and variants (syn, synner) yielded no viable results, including in contexts like The Canterbury Tales (Tale of Melibee, Parson's Tale) where both sin and sinner are used. I suspect the phrasing was not extant in Middle English.
After this, I used Early English Books Online, a corpus of pre-1700 printed texts, to see what I could find. To help narrow the results, I searched for "love the sinner" with sin as a collocation as well as "hate the sin" with sinner as a collocation. I mostly turned up printed sermons as results, and I sifted through them to find the results that worked.
Searching for "Love the Sinner"
In The Certainty of the Future Judgment (1684), Matthew Bryan asserts in a sermon the following:
Be∣lieve it, 'tis the sin we hate, while we reprove, but yet love the Sinner.
Finding the origin of the phrase in sermons seems promising, as sermons often use formulaic language to drive a point home, and the sin/sinner repetition is quite a formula. Early English Books Online turns up one earlier example of love the sinner where sin is also in the mix:
is the love of God first to man, which love being received, as
in the Light it is manifested, then man comes to be drawn with
the love of God, to love God again, and this is perfect love,
with which love the sinner is sought, and for sin reproved, where∣in
the Lord God doth make known his own compassion to a sin∣ner,
and doth manifest, That he would not the death of a sinner,
but hath given his onely begotten Son, that whosoever believeth in
him should not perish but have everlasting Life. (The Standing Truth, William Smith, 1663)
Searching for "Hate the Sin"
In The Soules Progresse to the Celestiall Canaan, or Heavenly Jerusalem (1639) by John Welles of Beccles, he makes the earliest appeal to hating the sin but doing something with the sinner that I can find:
if in our friends wee ought to admonish them, and tell them of their fault; if in our enemies, we must hate the sin, but pity the sinner, and labour if it be possible, his conversion, but not his imi∣tation
Meanwhile, in Paraphrastical Meditations Upon Isaiah 55 & Psalm 51 (1649) by John Barford, he has something very close to the target phrasing:
Wee must hate the sin but not the sinner.
And in the same year, the same phrasing appears in A Clavis to the Bible (1649) by John Trapp, while Trapp is commenting on Deuteronomy 25:
There is an honour due to all men, 1 Pet. 2.17. and though we must hate the sin, yet not the sinner.
The phrasing begins to show up in the Early English Books Online corpus by 1639, and there are several results in a similar spirit up to the end of the corpus in 1700. Based on these results, I would guess that hate the sin, love the sinner or similar phrasing was established rhetoric by the middle of the century. Because religious rhetoric of the era was often spoken, the phrasing possibly had a history of being spoken for decades before the first attestation in print.