One thing is for sure: they cannot be redacted from their character, under any circumstance. This is just not an option if you want to write correct French. What’s more, if you omit them, you might introduce ambiguities, because those diacritics sometimes distinguish important variations.
Diacritics are part of French orthography. To take one example, “dû” is the past participle of “devoir”. If you remove the circumflex, it becomes “du”, the contraction of “de” + “le”.
Different diacritics denote different things. The circumflex over an vowel often marks where an “s” used to follow the vowel. So modern French “êtes” is “estes” in old French. The “s” in the middle was dropped, and a circumflex placed above the “e”. English speakers can mentally insert an “s” after a circumflex and guess at a word:
- forêt => forest
- hôpital => hospital
- tempête => tempest
Accent aigu changes the pronunciation. “parle” as in “elle parle” (“she talks”) has one syllable and rhymes with the first name “Carl”. But “parlé” as in “elle a parlé” (“she talked” or “she has talked”) has two syllables; the first rhymes with “Carl”, the second with “hay”, like “parlay” (and yes, that’s a cognate).
The cedilla makes a “c” soft where it would otherwise be hard, so “français” has a soft “c” in the middle, like an “s”.
Search Wikipedia and you can learn about other diacritics and other uses for the diacritics I mentioned.
It is acceptable to omit diacritics from capitalized letters, but not from lowercase ones.
As noted in the comments, while diacritics are often omitted from capital letters, this is considered non-standard by the Académie Française for precisely the reasons listed above: diacritics are part of French orthography, and removing them leads to ambiguity.
For reference, the usual diacritics are as follows.
Accent aigu: é
Pronunciation: Uniformly causes the vowel to be pronounced [e] (as in English “may”).
There are some rare exceptions where it’s pronounced [ɛ] (as in English “beg”) instead, such as événement (recommended spelling changed to évènement in 1990) and in inversions like aimé-je.
Usage: Most notably on past participles of -er verbs, such as parlé. Many nouns end in it as well, such as musée. Besides that, it’s simply to mark the “ay” sound in many words, such as présenter. One word that shows both cases is the surprising créé (past tense of créer).
Notes: An e without an accent sometimes disappears from pronunciation (the e muet). However, if it has any accent, this is not possible. An e with an accent other than é is generally pronounced [ɛ].
Pronunciation: Causes a hard c [k] to be pronounced as a soft c [s] instead.
Usage and notes: See this answer.
Tréma: ä ë ï ö ü ÿ
Pronunciation: Pronounce this and the preceding vowel as two distinct sounds.
Usage: Purely used in fixed forms, not triggered by any grammatical shift. Some examples are aïeux, ambiguë, and borrowed words from languages that use this more often, such as Hawaï.
Notes: ï is by far the most common, followed by ë.
Accent grave: à è ù
Pronunciation: This accent on à and ù does not change their pronunciation. However, è is pronounced [ɛ].
Usage: à and ù are only in fixed forms, such as là “there” vs. la “the” and où “where” vs. ou “or”. However, è can appear not only in fixed forms, such as the common ending -ère (père, prière, bière), but also grammatically.
Specifically, it appears when an e that is normally caduc becomes obligatoire as a result of conjugation. For example, in the verb lever, the first vowel is reduced or even dropped. But in the present-tense conjugation lève, which has only one syllable, the stress has moved back to the first e and it is no longer optional. This is marked with the accent grave. (This is not 100% regular.)
Accent circonflexe: â, ê, î, ô, û
Pronunciation: The vowel usually becomes the variant it would have in an open/”long” syllable. The difference is negligible for î and û. For ê, the same note as above applies: if it has an accent, this letter can’t be silent. For â, it becomes a back vowel [ɑ]; compare sache and lâche.
Usage: A few different purposes. One is to distinguish words, e.g. du “of the” vs. dû “had to” or jeune “young” vs. jeûne “a fast from food”. Another is to signal historical letters that have disappeared, often s, as in fête, honnête, forêt (compare feast, honest, forest). It also regularly appears in certain conjugations, such as the passé simple and the past subjunctive.
Capital letters are often written without their accents. This leads to some amusing situations, like this sign I saw at the Eiffel Tower advertising some seemingly unappealing snacks:
(salé “salted” vs. sale “dirty”)
Edit: As jlliagre and Frank point out, although it is common practice, omitting accents on capital letters is not considered correct French by the Académie.
As @btrem and Luke Sawczak have said in relation to ê, the circumflex placed above an e often denotes an obsolete adjacent s, as in forêt, hôpital and tempête. Similarly, é at the beginning of a word often also denotes an s found in English words: élève (slave), école (school), étudier (study) etc.