I am trying to do the evolution from Classical Latin to Vulgar Latin of this word: Lanius non laneo. Could someone help me? What are the changes that occur? I was thinking of a diphthongation but I'm not sure.
Diphthongization is not involved. We can identify two separate differences between "lanius" and "laneo": the use of "ni" vs. "ne" and the use of the ending "-us" vs. "-o".
Confusion between the spellings "ni" and "ne" is the result of unstressed prevocalic /i/ and /e/ merging in pronunciation as a non-syllabic palatal glide [j] (eventually, this [j] affected the pronunciation of the preceding consonant or merged with it). That sound change occurred as early as the first century, as its effects are mentioned already in the extracts of Cornutus that are passed down by Cassiodorus (Cornutus says "vineas" and "vinias" are nothing more than alternative spellings of the same word, and criticizes attempts to distinguish them by meaning). So it seems likely that for the author of the Appendix Probi, the difference between "ni" and "ne" would be a matter of spelling alone (like the difference in modern English between the correct spelling "extraneous" and the incorrect spelling "extranious").
The difference between "-us" and "-o" likely refers to the use of the second-declension lanius, lanii vs. the third-declension lanio, lanionis. In Late Latin, it seems to have become popular to attach the ending -ō, -ōnis to some nouns: compare Classical Latin talus to Italian tallone or French talon. The author of the Appendix Probi may have preferred the version of this word without the suffix.
The concept of "Vulgar Latin" may not be very useful, and the Appendix Probi doesn't necessarily give us a view of a distinct "Vulgar Latin" language any more than the lists you can find of "commonly misspelled/mispronounced English words", with entries like "mischievous not mischievious", give us a view into a separate "Vulgar English" version of the language.
There are several factors in this morphological shift which is based on verbal mispronounciations:
(1) mistaking the vowel -e (ay) for -i (ee), a very common shift
(2) Loss of Case System: classical Latin had a complex system of noun cases, which became simplified in Vulgar Latin and eventually lost in Romance languages. In Italian, the nominative case endings (which often included -us for masculine nouns and -ius for many occupations) were replaced by the simpler -o, which ultimately derived from the Latin accusative case ending (see below) that became the standard form for the nominative case in Italian and other Romance languages.
(3) Simplification of Endings: The trend towards simplified declensions and conjugations made it easier for speakers to create and understand occupational names. The -o ending in Italian occupational titles is consistent with the masculine singular noun ending in the language, which makes it a natural fit for nouns denoting male professionals.
(4) The reason why the Latin nominative -us changed to -o in Italian is that first the nominative coalesced with the accusative case, so that words ending in the nominative -us began to be pronounced indifferently as though they were accusative (-um), so we might have the vulgur pronounciation laneum, for example, instead of lanius. Then, the -m began to be dropped and the short U-sound ("uh") gets mispronounced as a short O ("awe"), so laneum becomes laneo. Eventually in modern Italian the short O (awe) became long (owe).