One of the French language difficulties is the significant number of letters, often consonants but also vowels, mostly E, that are either never pronounced or optionally pronounced.
For example the final S of bois is always silent when bois is singular, ending a sentence, or followed by a consonant.
The final E of de is what is called a schwa [ə], an e muet (mute e) or better, a e caduc, i.e. an E that can fall, i.e. can be realized as [ə] or stay silent [Ø].
Dropping the E of de is mandatory when it is followed by a word starting with a vowel and in that case the E is replaced by an apostrophe, e.g.:
un feu d'artifice.
With feu de bois, the case is close but while this E is dropped by a large majority of French locutors, an apostrophe (diacritic used since the Greek and Latin to represent an élision) is never used in academic French to show this drop. This might however be done when representing spoken French a relaxed way giving:
The first syllable is unchanged while the second one aggregates the former second and third ones. I have never seen this represented as feud bois because if you force a pause to occur in the middle of the phrase, it will happen before the D.
Many french words are ended by an E caduc which, depending on the words, is very often or always dropped outside Southern France. e.g.:
Il est arrivé l'premier.
When you have a sequence of syllables each one containing an E caduc, some variations can be heard depending on which ones are dropped:
Feu de cheminée (5 syllables)
Feu d'cheminée (4 syllables)
Feu d'ch'minée (3 syllables)
Je te le dis pas (5 s)
might be pronounced as one of:
Je t'le dis pas (4 s)
Ch'teul dis pas (3 s)
Chtleu dis pas (3 s)
Other vowels might also fall the same way but this is often stigmatized:
C'est elle qu'a raison (for qui a raison)
Finally, while it is true that much more e caducs are dropped in "Parisian French" than in southern France, there are a few cases where an E is kept in both Paris and Marseilles but often or always dropped in the Franco-Provençal (Arpitan) area like Genève pronounced J'nève there.
Regarding whether to pronounce them or both, there is probably zero risk to be misunderstood if you pronounce all the optional E's and a very small one if you drop too much of them.
After all, we understand three years old kids who add a pseudo liaison and say:
J'ai vu le navion.
J'ai vu l'avion.
How do you know when the /ə/ drops and when it doesn't?
Question on the deletion of schwa
«L’Est-e du Québec», «un film-e d’horreur»: des E euphoniques?