There is also a paucity of examples of common Greek words from antiquity that began with iota eta. Many of these words are only known to us because they appear in Hesychius of Alexandria's "Alphabetical Collection of All Words" (Συναγωγὴ Πασῶν Λεξέων κατὰ Στοιχεῖον), includes approximately 2640 entries, a copious list of peculiar words, forms and phrases, with an explanation of their meaning, and often with a reference to the author who used them or to the district of Greece where they were current. I have highlighted in red the Hesychius entries and regional variants of words that usually began with ἰα, ἰο or another vowel:
ἰή LSJ, Middle Liddell, Slater 245 0 (= exclam. of joy, enthusiasm or grief)
ἰή (2) LSJ 245 (var. of ἰά)
ἰηγορεῖν LSJ 0 0 joy (= Hesychius)
ἰήϊος LSJ 0 0 (= exclam. invoked with the cry)
ἰήιος Middle Liddell 9 9 (= exclaim. invoked with the cry)
ἴηλα LSJ, Autenrieth 0 0 (= Hesyschius, Ionian var. for ἰαλ-)
ἴημα LSJ 0 0 (= Ion. for ἴαμα)
ἵημι LSJ, Middle Liddell, Slater, Autenrieth 62,925 521 (v.l. ἵεις S.El.596, Castorio 2), ἵησι, 3pl.
ἱᾶσι, Ion. and Ep. ἱεῖσι; imper. “ἵει” Il.21.338, E.El.593 lyr.)
ἴηνα LSJ 0 0 (= aor. 1 Act. of ἰαίνω)
Ἰηπαιήων LSJ, Middle Liddell 2 2 hymn (= Ἰηπαι-ήων , ονος, ὁ, epith. of Apollo, from the cry ἰὴ παιῆον, h.Ap. 272)
Ἰηπαιωνίζω LSJ 0 0 cry
ἰηπαιωνίζω Middle Liddell 1 1 to cry
ἰήρια LSJ 0 0 (τά,= ἰατήρια, dub. in Supp.Epigr.1.414.4 Crete, v/iv B.C.)
ἰήσασθε Autenrieth 0 0 (= see ι?α?ομαι)
ἴῃσι LSJ, Autenrieth (= Ep. 3sg. pres. subj. of εἶμι (A.ibo). ἰήσιμος , ἴησις , Ion. for ἰασ-. ibo )
Ἰησονίδης Autenrieth 0 0 son of Iēson (Jason), Euneus, Il. 7.468, 471, Il. 23.747.
Ἰησοῦς LSJ, Middle Liddell 0 0 Joshua
Ἰήσων Autenrieth 0 0 Iēson (Jason), the leader of the Argonauts, Od. 12.72
ἰήτειρα LSJ (ἰητέον , ἰητήρ , ἰητόριον , ἰητρός , etc., Ion. for ἰατ-).
ἰήτης LSJ 0 0 (τοξότης, ἰοβόλος, Hsch.)
ἰητρός Autenrieth 0 0 (ἰα?ομαι: healer, surgeon, physician; with ἀνήρ, Il. 11.514)
It would seem to me at least that the only naturally occurring examples of ἰη connect it with a cry or exclamation. The most obvious example is the term Ἰηπαιήων in the hymns of Apollo.
This term derives its origin from Παιάν, an appellation of Apollo as the healing deity ; the burden of the song being Ιη or ίώ ΙΙαιάν, in thanksgiving for deliverance from evil. Compare the paean of Aristonous (Smyth Melic Poets p. 527), with the repeated formulae ἰὴ ἰὲ Παιάν, ὦ ἰὲ Παιάν, Timotheus - Aesch. Pers. 218; the latter (fr. 25 Wilamowitz) has also ἵε παιάν, the aspirate being due to the supposed connexion with ἵημι (βέλος), for which see Athen. 701 C. With the origin of the word from this refrain cf. the similar history of the Linus-song, the hymenaeus, and the iobacchus; the last, like Ἰηπαιήων, was a title of the god, as well as the name for the hymn. On Παιάν and Παιών see Preller-Robert i. p. 241 n. 2, p. 277 n. 2, Pauly-Wissowa Apollon 62, Smyth Melic Poets p. xxxvi f., and further on 500. Macrobius's important passage on this cry is here.