I don't know of any word where "pph" has been replaced with "ph". The fact that we only have "sapphire" and "Sappho," both starting with "sa-" (and of course related words and names such as Sapphic/sapphic, Sapphira), seems to be a coincidence.
It seems this was a rare sequence of sounds in Greek, like the "tth" of "Matthew":
Neither occurs (as far as I can think) in any native Greek words, only in borrowings. – Janus Bahs Jacquet May 24 at 10:09
In Greek, the π is not silent. In modern Greek, the sequence "πφ" would be pronounced /pf/; you can hear this in JovannaK's pronunciation of Σαπφώ on Forvo. Ancient Greek is thought to have had aspirated consonants and doubled or long ("geminate") consonants in some contexts; the ancient pronunciation of "πφ" is, as Janus says, reconstructed as something like /p.pʰ/, distinct from "singleton" φ /pʰ/. (The pronunciation of peri22b on Forvo seems to be aimed towards this /p.pʰ/ realization.)
I'm not an expert, but my impression is that in Ancient Greek, aspiration contrasts were mostly neutralized for plosives that came before other obstruents. The neutralized sound could be written in various ways, some of which were probably conventional; e.g. when the second consonant was an aspirated plosive, the first seems to have also generally been written with a letter for an aspirated plosive (hence the common occurence in English words of Greek origin of sequences like "phth" and "chth"); when the second consonant was the fricative /s/, the special letters ψ and ξ were used, and so on. It seems that /p.pʰ/ and /t.tʰ/ were usually written as πφ and τθ respectively but the alternative spellings φφ and θθ were also possible in some eras (History of the Greek Alphabet,
by Evangelinus Apostolides Sophocles, 1848); maybe they represented a difference in pronunciation as well or maybe not.
In Greek, generally the letters Φ ("ph"), θ ("th") and χ ("ch") were used to transcribe Hebrew ף ("p/f"), ת ("t/th"), and ך ("k/kh") respectively, or the corresponding sounds in a similar Semitic language such as Aramaic. (this is a simplification, but I can't give a more nuanced description because I'm not an expert)
(Side note about something that can cause a bit of confusion: Hebrew has rules assigning a fricative pronunciation to these some of these letters in certain contexts, and a plosive pronunciation in other contexts; the precise rules are a bit different and apply to different letters in different varieties of Hebrew. The Greek transliteration generally is not at all related to this phenomenon in Hebrew: the letters Φ θ χ may correspond to Hebrew plosives as well as Hebrew fricatives. The explanation I've seen for the Greek transcription system is that even the plosive allophones of the corresponding Hebrew consonants were aspirated, making them match better to Greek Φ θ χ than Greek π τ κ. Greek τ κ were the standard equivalents to Hebrew ט ("t", formerly contrastively "emphatic" compared to ת, and probably unaspirated) and ק ("q/k", formerly uvular or emphatic, and probably unaspirated). See the answer on Linguistics SE to Why was the name תאומא transliterated as Θωμᾶς (Thomas) rather than Τωμᾶς (Tomas)?)
The sequence "πφ" was used in Greek to represent geminate or double-length Hebrew/Aramaic ף, "pp". But there are not many words with this sound that were loaned from Hebrew/Semitic to Greek, and then brought into English. Sapphire is one: Wiktionary says
Borrowing from Old French saphir, from Latin sapphir, sappir, sapphīrus, from Ancient Greek σάπφειρος (sáppheiros, “precious stone, gem”), from a Semitic language (compare Hebrew סַפִּיר (sappī́r)
Another example, that is an obscure name, is "Jonathan Apphus" (Wikipedia says "Hebrew: יונתן הוופסי, Ancient Greek: Ἰωνάθαν Ἀπφοῦς"; "JONATHAN MACCABEUS" on Jewish Encyclopedia says "Syriac הפּוש" which looks a bit closer to "Ἀπφοῦς" going by the little I know of the Hebrew alphabet).
I have no idea where "Sappho" comes from, but as you say, "there appears to be no etymological link" so it seems extremely doubtful that the similarity to the start of "sapphire" is anything but a coincidence. (If anyone knows, please post the answer to the following Linguistics SE post: Two questions about Sappho's name.)
The modern English pronunciation can probably be explained by considerations within the sound system and orthographic system of English (the sound sequence /pf/ is awkward, and in many cases doubled letters are pronounced the same as single letters in English, so there's no strong pressure from the orthography to use /pf/ rather than /f/).