Dictionaries

They vary:

• The TLFi presents the [œ] pronunciation first, but also gives a list of the choices made by various dictionaries in the past, including:

  • Spelling as déjeûner (to make it consistent with the [ø] pronunciation) in some dictionaries from the XVIII^th & XIX^th Centuries, including the Dictionnaire de l’Académie française of 1740 and 1762 (earlier version, 1694 and 1718, had desjeuner, and later releases had the modern version déjeuner).
  • Pronunciation with [ø] for some dictionaries, either exclusive or as a possible variant of [œ].
  • There’s even the possibility of skipping the vowel altogether mentioned in a 1935 dictionary: [deʒne], in which the sound drifted initially to [ə] before being treated as a silent e.

• Le Petit Robert has exclusively indicated the pronunciation [œ], for at least the 1980 to 2010 period. But the same publisher, in another dictionary, Le Dictionnaire culturel en langue française, published in 2005, indicated only [ø].

• Hachette indicates [œ].

Usage

It varies as well. My pronunciation is pretty close to /deʒəne/, that of currently the only other user to have answered here is /deʒøne/. That is statistically insignificant, but it seems to be a reduced version of the diversity one would find if randomely asking French-speakers about their personal pronunciation.

Indeed, to add to the variety already shown in the dictionaries, there is an inquiry about oral French, for which data acquisition is still taking place at this time (June 2109), but whose preliminary results shown on this page are already quite clear:

• The distinction between /ø/ et /œ/ doesn’t take a fixed form and varies by region;
• Its variations can be mapped surprisingly clearly, here is an example of a pair of words, one of which is rather closely related to the question.

This simple picture using a two-colour scale might make it appear like a dipole where two linguistic habits are delimited, but reality is actually more complex. Two comments made below this answer hint at how diverse the local realities could be:

(About the lower blue part of the map) It’s not that there is no distinction between [ø] and [œ] in this area, but rather that there is no distinction between /ø/ and /œ/, as their realization is regularized (similarly to the other vowels) to [ø] in open syllables and [œ] in closed ones. Which means jeune and jeûne use [œ] while déjeuner uses [ø].
   _{—Stéphane Gimenez}

Belgium usually shows the contrast, but that pair is consistently realised [ʒœːn] or [ʒœ̃ːn] (and déjeuner has an /œ/ too), while the blue part of Switzerland has both as /ʒøn/.
   _{—Eau qui dort}

The Dictionnaire culturel en langue française, cited earlier, also have a special mention about the pair of phonems [œ] and [ø] at the beginning, where they present the pronunciation standards for the dictionary as a whole. It states that there are (as of 2005) two diverging tendancies at work in spoken French regarding these two phonems:

• A first tendancy would stick to one sound for any specific root, making sure pairs like aveugler/aveugle agree (the default presented is /œ/);
• A second tendancy where the position of the sound in the final syllable of the word would close the vowel (making the pair /avœɡle/ and /avøɡl/).

This can actually be used to further show how usage vary in the real world. To my French Canadian ears, all this sounded very strange, since the vast majority of Quebecquers will pronounce aveugler and all its conjugated forms as /avøɡle/, while we have a split lot over aveugle adjective or noun (I feel /avøɡl/ was more present up to the 80’s, before it started eroding away as /avœɡl/ was gaining popularity, but though the current state of affair might be transitional, both pronunciations are currently present).

Conclusion

It looks like there is no consensus, and therefore no right or wrong answer to this.

The best bet for a native speaker is to go with what they are used to or what they feel comfortable with given the immediate social environment.

For learners of the French language, it’s probably best to try to replicate the way it is pronouced by locals if on a visit, or by their teachers and/or francophone contacts otherwise.

As the TLFi gives /ʒœn/ for “jeune” and /ʒøne/ for “jeuner” and that I agree with that after having heard the vowels from the wiktionary, as the TLFi also gives /deʒøne/ for “déjeuner” with which my pronunciation also agrees, I think you should say /deʒøne/; however, it is quite possible to find /deʒœne/, that’s the pronunciation of certain people in certain régions, although not necessarily of all; for instance this usage is asserted by the TLFi for “heureuse” : [øʀø], fem. [øʀø:z] ou [œ].