Latin makes use of macrons (small lines above letters) to indicate a different pronunciation for that letter.
I'm particularly interested in the latter - while I have a general idea of the impact a macron has on a letter, I'm unsure whether it has any wider impacts.
In most modern texts, the purpose of using macrons (or macra) is to indicate pronunciation, so they're usually pretty straightforward. (The use of macrons in this manner was not a feature of Classical Latin, although vowel length could be marked unsystematically in Classical Latin by certain other devices, such as the apex and "i longa", a taller version of the letter I.)
It's true that macrons can indicate several aspects of pronunciation.
A macron usually indicates that the modified vowel is long. In the context of Latin, "long vowel" should be understood quite literally: keep making the vowel sound until a longer period of time has passed than it would take to pronounce a short vowel. Vowel length was what is called phonemic in Classical Latin, which means that there were pairs of words for which the only difference in pronunciation was the length of some vowel: for example, liber "book" with short i vs. līber "free" with long ī.
At one point, duration may have been the only difference between long and short vowels. In the International Phonetic Alphabet, we transcribe vowel length with a triangular colon ː after the vowel; so we could say that the early Classical Latin pronunciation of the vowels we spell a ā e ē i ī o ō u ū would be /a aː e eː i iː o oː u uː/. We don't know for sure the precise phonetic quality of these vowels. Andrea Calabrese has proposed that they were [ɑ ɑː ɛ ɛː i iː ɔ ɔː u uː] (On the Evolution of the Short High Vowels of Latin into Romance).
It is known that at some point during Classical Latin or between Classical and Vulgar Latin, most pairs of long and short vowels also became distinguished in quality. (Calabrese mentions Allen and Sturtevant as linguists who reconstruct these quality distinctions back to the Classical Latin stage.) The non-low vowels (all but a aː) clearly show distinct qualities for short vs. long in some of the Romance languages descended from Vulgar Latin, which causes us to reconstruct something like [a aː ɛ eː ɪ iː ɔ oː ʊ uː]. I believe I've read some accounts that also reconstruct a difference in quality for short a and long a along the lines of [ɐ aː], but I don't know what evidence this is based on. Short unstressed i between a consonant and a vowel (which in Romance often developed to some kind of glide, or palatalized the preceding consonant and was lost entirely) may have had a slightly different quality more like [i] or [ij], at least for some speakers or some of the time.
Vulgar Latin is also believed to have had long [ɛː], but as the realization of ae rather than of ē.
A warning: sometimes when dealing with poetry or old dictionaries, you might see macrons used (or misused) just to mark heavy syllables. Lewis and Short is a particularly misleading resource in this regard: don't trust it to tell you when vowels are long, as it tends to use the macron instead to mark the scansion of the syllable.
In Latin, a syllable is "heavy" if it has a long vowel, or ends in a consonant, or both. Heavy syllables are relevant to rules of word stress and poetic scansion.
Syllables that end in a consonant are always heavy in poetry, but they may contain either long or short vowels. There is a confusing old tradition of referring to a short vowel in such syllables as being "long by position", but they aren't necessarily actual long vowels.
In fact, the vowel in such syllables has a "hidden quantity," so be careful in situations like this and try to check several sources that distinguish between short and long in closed syllables to see if they agree on the length of the vowel. We cannot reconstruct vowel length here based on metrical evidence, but we sometimes can based on inscriptions that use the apex, vowel quality in descended Romance words, or descriptions of ancient grammarians.
It's usually fairly obvious if a syllable ends in a consonant in Latin, but there are a few tricky cases where a single letter may represent a double consonant (which is always divided between two syllables for metrical purposes). In particular, the consonants j and z are not written double, but generally represent double consonants in Latin, so a vowel preceding either of these will have hidden quantity (and might be written with a macron even if the vowel is short, to indicate that the syllable is heavy). For example, the word most dictionaries write as māiŏr was, according to the linguist W. Sidney Allen, actually pronounced as /majjor/ with short "a," but a long /jj/. See the question When is an I not an I?
There are also some cases where it seems word-final consonants in Latin were pronounced long, and you may see this written as a macron on the preceding vowel. According to The Latin Language: A Historical Outline of Its Sounds, Inflections, and Syntax, by Charles Edwin Bennett (1907), a long final consonant may have occurred in some words where original -s was assimilated to a preceding s, l, or r (as in far, farris, possibly pronounced /farr, farris/). Another word that is believed to have had a long final consonant is the nominative/accusative singular neuter pronoun hoc (sometimes written hōc) which Bennett says was pronounced /hokk/ and not /hoːk/. (The masculine and neuter ablative form hōc is reconstructed with a genuine long vowel.)
Another related point of pronunciation: macrons are generally always used on vowels that occur in the combinations Vnf and Vns (where "V" stands for any vowel). Although this means there is no contrast here between short and long, it does appear that these vowels were phonetically long (The Latin Language: A Historical Outline of Its Sounds, Inflections, and Syntax, by Charles Edwin Bennett). In languages descended from Latin, the consonantal /n/ sound was actually regularly dropped from Vns clusters (example: Italian isola < Latin īnsula). However, when "ns" occurred due to a prefix ending in "n" coming before a stem starting with "s", analogical retention or restoration of /n/ in Romance languages is common. And since pretty much all instances of "nf" in Latin are due to prefixation, I don't actually know of any word in a Romance language that has denasalized a Latin "nf" cluster.
I don't remember if there is any consensus on if a nasal consonant was phonetically present in these sequences in Latin. In any case, it is believed that in classical times the preceding vowel in such words was pronounced with nasalization, so sequences like īnf would still have contrasted with sequences like īf in Classical Latin even if the consonant /n/ was already elided: it would be [ı̃ːf] vs. [iːf]. (If a nasal consonant was present, it was likely assimilated to the labiodental articulation of the following /f/, giving the pronunciation [ı̃ːɱf].)
As far as I know, a macron doesn't affect the pronunciation of any other sound directly. But knowing that a vowel is long may help you apply the regular Latin stress rule, whereby a word is stressed on the second-to-last syllable if it is heavy (contains a long vowel or diphthong or ends in a consonant) and on the third-to-last if the second-to-last is light (contains a short vowel and doesn't end with a consonant). In addition, as jwodder pointed out in a comment, a macron on either vowel in a sequence like ae, oe, au, eu tells you that the two vowels are pronounced separately rather than as a diphthong.
It’s not really right to say that “Latin” makes use of macrons – most Latin texts are written without macrons. Rather, some specific types of text (dictionaries, grammars, textbooks for beginners, etc.) use the macron to indicate that a vowel is long; i.e. the sound is held for a longer time than in a short vowel. Similar texts for other languages often use similar auxiliary symbols; they may, e.g., indicate the stressed syllable by underlining.
In most languages that have a vowel length distinction, corresponding short and long vowels are not exactly identical; e.g., in most forms of English, the long [iː] (as in beat) is more open than the short [ɪ] (as in bit). It is likely that a similar difference existed in ancient Latin, and it most certainly exists in Latin as pronounced today by native speakers of such other languages.
What kind of other effects did you have in mind?
As far as I know, a macron merely indicates an increase in the length of the vowel, and nothing else. So the vowel should change in quantity only, not quality. Consonants around it should not be affected, nor any other syllables.
Concerning ecclesiastical pronunciation, Collins' Primer (pg. 4) says that vowel quantity determines the placement of the accent:
Accent in Latin is determined by the quantity of the next to last syllable( called the penult); if the penult is long, it bears the accent [...]. If the penult is short, the then the third syllable from the end (called the antepenult) gets the accent [...].
