Since you are asking about Latin and not Proto-Italic, my answer is going to address Latin only (so I will ignore e.g. the following sound change proposed by Philip Baldi, based on the glotttalic theory, (word-initially) PIE *bʰ > Proto-Latin *pʰ > Proto-Latin *ɸ > Latin f).
-mf- in Latin inscriptions
Such a bilabial hypothesis is indeed usually mentioned in many scholarly Latin grammars.
For instance, Manu Leumann (Lateinische Grammatik, 1977) argues that
"Inschriftlich -mf- deutet, trotz Hermann, NGG 1919, 249, auf ältere bilabiale Aussprache (wie bein p b)" (p. 173, see §177b). Some of the examples he finds relevant are below:
IM FRONTE (CIL I2 1420, line 5), COMFLVONT (CIL I2 584, line 13), imfo(s)sos (Kretschmer, Nehring, and Kroll 1925: 237); comficiendum etc.
cf. Sommer and Pfister 1977 (Handbuch der lateinische Laut- und Formenlehre):
"Für bilabiale Artikulation in republikanischer Zeit spricht, daß der vorhergehende homorgane Nasal einige Male als m, nicht, wie später die Regel, als n erscheint (p. 147, §112).
Note: It seems Sommer himself (Sommer 1914/1948) did not support the bilabial hypothesis, cf. "Gegen bilabiale Artikulation Sommer2". The 1977 edition was revised by Raimund Pfister.
cf. Niedermann 1945 (§7, note 2); this note is absent in the later, 4th ed. (Niedermann 1953).
Sturtevant 1920 is overly optimistic in my opinion – cf. his “Since m was a bilabial consonant, f also must have been bilabial in early times” (p. 91).
Tronskij 1960 is more cautious:
«В прошлом, возможно, имел губогубной характер» [It could have been biliabial] (Tronskij 1960: 54, §74).
cf. Lindsay 1894 (I highly recommend reading §114, pp. 98-100):
"Their [i.e. "of the Latin phoneticians" - Alex B.] words leave no doubt whatever that f was a labiodental spirant" (p. 98) [...] "But it is highly probable that Latin f was at some time bilabial" (p. 99), see p.99 for his argumentation.
However, Jan Safarewicz (Historische lateinische Grammatik, 1969) does not find it compelling, cf.
"Doch ist diese Hypothese nicht zwingend, denn eine solche Assimilation konnte auch vor einem labiodentalen f auftreten, erscheint sie doch in der Tat öfter in späteren Zeiten, als die Artikulation bestimmt labiodental war" (pp. 27-28),
cf. Allen 1978: 35 (Vox Latina, 2nd ed.), who essentially repeats what Safarewicz wrote, without mentioning him though:
"But even if such examples were common, the evidence would be inconclusive; for the preceding nasal can well have had a labio-dental articulation (cf. p. 29), and it is then purely a matter of orthographic convention whether it is represented by the sign of the normally bilabial m or the normally dental n."
cf. Stuart-Smith 2004: 47:
"these spellings could equally be interpreted as attempts to represent a labiodental nasal [ɱ], which would not be unexpected before [f]." (also see p. 46).
Latin f > Old Spanish h
(word-initially, except before liquids and w)
Penny 2002 writes that “The process by which Latin F- came to be eliminated from most popular Spanish words has been the subject of intense debate” (p. 91). He mentions two accounts, the substratum influence (possibly Basque) vs. “purely intra-Latin explanations” (evidently, Ralph Penny himself).
NB: Tuten, Pato and Schwarzwald 2016 add that this change did not happen in Asturian or Aragonese, where Latin f was retained (p. 389).
About the second proposal, Penny says that “Those who seek an internal Latin explanation for the loss of F- in Spanish have sometimes turned to regional Latin pronunciations” (p. 92).
And who are those scholars exactly? Well, Penny only mentions his own work (Penny 1972a and Penny1990b).
What is his argumentation?
Penny writes that “It is possible that the spoken Latin of remoter areas (such as Cantabria) preserved a bilabial articulation (/ɸ/) of F-, which had earlier been normal in Latin but which had been replaced by labiodental /f/ in Rome and in those areas in closest contact with Rome” (p. 92). What is his evidence for such a claim? ...
Alkire and Rosen 2010 are more cautious and merely say that this development, Latin f > Old Spanish h, is "of uncertain origin" (p. 50).
Was Spanish Latin that archaic?
Now, if you need to gain a deeper understanding of the data pertaining to the regional diversification of Latin, the magisterial tome by J.N. Adams (Adams 2007) is the book that cannot be ignored – or, as they sometimes say in blurbs, “a scholar can ignore it at their peril.”
He delivers a fatal blow to the widely held view that Spanish Latin was archaic, examining all the relevant data with utmost precision and unparalleled critical analysis. I strongly recommend reading Chapter VI “Spain” (pp. 370-431), especially 6.2 The supposed conservatism of Spanish Latin.
Some of his points are:
- The supposed archaisms do not occur in the Latin of Spain “but in
early texts by writers who were not Spanish” (p. 399);
- The research of the supposed archaic Spanish Latin “has not always
been pursued with proper rigor” (p. 401), and Adams carefully
examines all the relevant data. He concludes his chapter by saying that some of the words argued to be archaisms must be disregarded because they do not withstand critical analysis, the others are of arguable relevance. For instance, re: fabulor (a word mentioned by Ralph Penny in the linked book), he says it
is “too common in the Empire […] to justify the conclusion that
reflexes in Spain represent an archaic survival from the early
Republic. This is merely a possibility” (p. 385) etc.
(I'll add more on Pulgram 1979 later)