This is a matter both of regional habits and general habits. In the south of France there’ll be people that never will drop the e in “feu de bois”; on the other hand in the region of Paris people have a different accent, in which certain e’s are dropped, and they’ll say regularly “feud bois”; nevertheless, Parisians will also say “feu de bois” in particular cases, as for instance when they don’t understand what is being said and repeat the words using the full pronunciation. Therefore you must be familiar with both pronunciations.
This is just as in English in the case of for instance the word “naturally”; you can say “natsh- er-el-i” or something like that but most of the time you say “natsh-e-li”. There is then a third type of usage which is characterized by a rather indifferent use of the two pronunciations: you can say “feud bois” some of the time and also for some of the time “feu de bois”, as when speaking in a more leasurely manner; people who make use of the two pronunciations are of that sort with parents from different regions, or people that have lived in two regions, or people whose accent is a more neutral accent (Loire region), neither from the South nor from Paris.

Whether you should or not drop the e in locutions such as “feu de bois” and in other contexts where the e is regularly dropped (Il est tombé dans l’feu (le feu) ? Il est tombé d’dans (dedans) ?) is not a question anyone can answer. You’ll have to make a choice. Nevertheless, you should know that wanting to attain to a great versatility and for example try to be fluent in the accent from the south of France and at the same time in an accent such as the Parisian one or the northern accent will cost you a lot of time, and it is an attainment that is practically never reached by the French themselves; it is deemed rather useless.

My opinion is that you should in any case work so as to develop habits in one given accent first so as to have a reference accent, that you can feel confortable with; you probably won’t want to change it later, accents are difficult to change. You can choose it according to your fancy (either the accent from Paris, or the “accent du Midi” in which e’s are not often dropped or a more neutral accent, although if you are sure that you want to use your French nowhere else than in a country such as Belgium or Canada, you might then consider adopting the accent from this given country). Once you keep a given accent, the particular habits of that accent as concerns the dropping of e’s will gradually become yours. You should strive however to keep to a single accent, although that prescription does not concern wholly dropping e’s as, for some of them, they can be dropped or not by people who have neither a true Parisian accent nor a true southern accent.

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:

feu d’bois.

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.

Elle part

is pronounced

El part

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)

can be

Feu d’cheminée (4 syllables)

or

Feu d’ch’minée (3 syllables)

and

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.

instead of:

J’ai vu l’avion.

😉

See also:

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?

Hmm I find the other answers a bit too much complicated and ambiguous. Let’s be clear: it’s nothing but an accent (the accent of most of the French people maybe, but still), it’s only “correct” in spoken French and not everyone speaks like that, and you are not supposed to write this way (I don’t get the comparison with dropping the e in de before a word that starts with a vowel: it’s a rule, you must drop it in such a case, in written and spoken French).

That being said, yes you would probably sound more French (depends on which French we are talking about) saying feu d'bois than feu de bois, like saying j'suis instead of je suis, and dropping the ne when using a negation (je suis pas d'accord instead of je ne suis pas d'accord), and many other things.

It doesn’t really work with many examples of X de Y, and I don’t think there is an ungrammatical rule that could help you know when it’s more acceptable. Fil d’pêche? Rather no. Argent d’poche? Rather yes.