An interesting query on Twitter read:
Apparently, Roman priests were not allowed to use iron razors or scissors… Anyone know why?
A little searching turned up some sources. I had hoped to find more in the old Realencyclopädie, but Bd. VI.2, col. 2489 gave only the same few. All are very late indeed.
My earliest source is from the 4th c. AD. It is Servius, Scholia on Vergil’s Aeneid, i, 448.
First the passage being commented on: Vergil, Aeneid, book 1, line.446-49:
446. Hīc templum Iūnōnī ingēns Sīdōnia Dīdō
condēbat, dōnīs opulentum et nūmine dīvae,
aerea cui gradibus surgēbant līmina nexaeque
aere trabēs, foribus cardō strīdēbat aēnīs.
Here Sidonian Dido was founding to Juno a mighty temple, rich in gifts and the presence of the goddess. Brazen was its threshold uprising on steps; bronze plates were its lintel beams, on doors of bronze creaked the hinges. (Loeb)
48. AEREA vel quod aes magis veteres in usu habebant, vel quod religioni apta est haec materies, denique flamen Dialis aereis cultris tondebatur: [aut quia vocalius ceteris metallis, aut quia medici aere quaedam vulnera curant, aut dicit quia veteres magis aere usi sunt] aut certe aerea saecula significantur: nam ut Hesiodus dicit, tempore quo haec gesta sunt aereum saeculum fuit. NEXAEQVE AERE TRABES multi ‘nixae’ legunt, non ‘nexae’, iuxta Varronem qui ait, Trisulcae fores, pessulis libratae, dehiscunt, graves atque in nixae in cardinum tardos turbines. Quidam trabes aeneas putant ipsum templum χαλκίοικον significari. Versus sane ipse hypermetros est.
BRONZE, or rather what was used as money by the ancients, or what was appropriate for religion, and then the Flamen Dialias was trimmed with a bronze knife: [or because more tuneful than other metals, or because doctors cured some wounds with bronze, or he says (this) because the ancients were more used to bronze] or at least the ages of bronze are signified: for, as Hesiod says, the time that this happened was the age of bronze. AND ITS ROOF-BEAMS WERE LINKED WITH BRONZE. Many read “heavy”, not “linked”, according to Varro who said, “The three-fold doors, from bolts released, they open, and in pushing on the hinge the slow heavy rotation.” Which beam Aeneas thought meant the temple itself was that “made of bronze” (i.e. of Athena). The verse obviously is in hypermeter.
Update (13/09/19): A kind commenter points out that I made rather a mess of this. His much better translation from here is:
AEREA, either because the ancients used bronze more widely, or because this material is appropriate for religious purposes; thus the hair of the Flamen Dialis was cut with bronze knives [or because it was more sonorous than other metals, or because doctors tended certain wounds with bronze, or he says that the ancients made more use of bronze] or at any rate the Bronze Ages are denoted: for, as Hesiod says, the time at which these events occurred was the Age of Bronze. NEXAEQVE AERE TRABES: Many read “resting upon” [nixae], not “plated with” [nexae], with Varro, who said, “The triple doors, released by bolts, open wide, heavy and resting upon slowly revolving pivots”. Some think that the “bronze lintel” means that the temple itself was a “shrine of bronze”. The line is obviously hypermetric.
In the 5th c. we have Macrobius, Saturnalia book 5, chapter 19, section 13:
Certainly there is much to show that it was commonly the custom to use instruments of bronze for sacred ceremonies, and especially in connection with rites whose purpose it was to entice or curse a person or, indeed, to drive out diseases.  I shall not comment on that line of Plautus:
“My chinking disease has its remedy-the chink of bronze,”
nor on Vergil’s reference elsewhere to:
“The ringing noise and sounding bronze of the Curetes” [Georgics 4. 15 I]
[ 13] but I shall quote the words of Carminius, a learned man and a most careful scholar, who says in the second Book of his work on Italy: “And so I find both that the Etruscans, in their sacred rites of Tages, were wont formerly to use a plowshare of bronze when they were founding a city, and that among the Sabines the priests used to cut their beards with rawrs of bronze.”  It would be tedious to seek to follow up these words of Carminius with a review of the many passages in which the most ancient of the Greeks habitually made use of the sound of bronze as being particularly efficacious. Let it be enough for the matter in hand to have shown that in introducing a reference to bronze sickles Vergil was following the example of a Greek author.
A note in the translation suggests that “Carminius” might in be a corruption for “Granius”. Macrobius is interesting, because, immediately before this passage, he discusses Sophocles and a similar Greek prohibition.
Finally in the 6th century we have John the Lydian, De Mensibus, i, 31, (tr. Mischa Hooker):
35. Under Numa, and before him, the priests of old times would have their hair cut with bronze scissors, but not with iron [scissors]. For iron, according to the Pythagoreans, is dedicated to matter: It too is dark and therefore nearly without form, wrought with much toil and useful for much, but not impassive.
The translator, Mischa Hooker, adds some valuable information in the footnotes:
90. Cf. Servius on Aeneid 1.448, with reference to the flamen Dialis; Macrobius, Saturnalia 5.19.11 (citing Carminius) mentions that Sabine priests had their hair cut with bronze. Iron has been the subject of taboos in various societies including Greek and Roman—see J. G. Frazer, Golden Bough, 3rd ed., 3: 225-36 (“Iron Tabooed”); for Greek religion more specifically, see the evidence collected by T. Wächter, Reinheitsvorschriften im griechischen Kult (Gießen, 1910), pp. 115-18; for Roman religion, note also Warde Fowler, The Religious Experience of the Roman People (London, 1922), pp. 32-35, 191, 214.
91. For iron as “dark” (melas), cf. Hesiod, Works and Days 151; “wrought with much toil” (polykmêtos) is a Homeric epithet for iron (e.g., Iliad 6.48). Proclus, while commenting on Hesiod’s “iron race,” explores the symbolism of iron as indicating earthliness, intractability, lack of rationality, and subjection to passions (Commentary on Plato’s Republic, 2: 77 Kroll). … further useful discussion of Greek religious restrictions on metals and rings appears in C. Le Roy, “Un règlement religieux au Létôon de Xanthos,” Revue Archéologique n.s. 2 (1986), pp. 286-9.
None of our three sources seem actually to know why there is a prohibition on the use of anything but bronze to cut hair. It is a defensible hypothesis that this was copied from the Greeks; or, equally possibly, that it arises spontaneously in these early prehistoric mediterranean populations, in reaction to the advent of iron. It would be possible to speculate endlessly.