As there is no official timeline to be found, fans of the series are left to theorize. What can be said however is that the original Breath of Fire is followed by Breath of Fire II and Breath of Fire III, this due to the second game being a direct sequel, and the third game having a mural under Dragnier depicting the fight against Myria. It is not however stated in which order the two of them comes, since Breath of Fire II takes place 500 years after the original and the age gap between the first and third game is not known.
The fourth and fifth games are more tricky to place, however, due to the thematically drastic differences in otherwise key recurring points. Some think they are not a part of the main timeline at all, but rather alternate universes.
Here is my take, note that I've only ever played the third and fourth games:
- Breath of Fire: Dragon Quarter
- Breath of Fire IV
- Breath of Fire
- Breath of Fire II
- Breath of Fire III
My reasoning behind placing the third game after the second is the, I assume, absence of III-occurring themes, thus meaning either those events were erased from existence or had yet to happen, of which I'm betting on the latter.
The reason IV is placed before the original is due to Yuna's experiments during the course of the game, creating near immortal beings. As he is alive after the ending, in addition to his comment of continuing his experiments, leads me to believe that he is the creator of Myria. As Deis stayed after the ending events (still without her body), it is possible that Yuna was able to alter her body in a matter similar to Myria, thus explaining Deis calling her "sister" in the end of Breath of Fire III. As of the Endless departing, well...
Dragon Quarter explains that dragons forced civilization underground by wreaking havoc on the earth. However, from what I've gathered without having actually played the game, it does not mention the complete annihilation of the dragon race. In addition, dragon hybrids still exist (Ryu's able to transform into one just like in III and IV), just like they do in other installments. The ending shows that the surface world is in no way destroyed, as well as having characters showing signs of being the precursors to the Grassrunner clan. Additionally, the Age of Machines is a time period described in III as being one of technological blossoming, something I think hints towards DQ.
There is a (that I can think of) flaw with this timeline, being that I am assuming that Brood and Endless are two separate entities, despite sharing many traits. This instead hints towards IV being set in another universe or planet, and thus another timeline entirely. Simply placing IV before DQ doesn't solve this issue.
However, if my theory regarding Yuna and Myria is correct, it should be noted that Elina is an Endless, not in that game recognized as a demon (but are so in the other installments). This would then mean that both Ryu and Fou-Lu in IV are not dragons at all, but rather demons. The similarity between Endless and Brood is therefore not coincidental, as seen within Breath of Fire III where both Deis and Myria in their respective forms share dragon-esque features (dragon being the loose term the Breath of Fire series uses to define anything reptilian), as well as both of them needing to transform into their respective forms, a trademark of the Brood. To further emphasize this point, Guardians Garr and Gaist are "anti-dragons" created by Myria, yet are both hauntingly similar to the beings they destroy, as well as, in the case of Gaist, are capable of transformation. Additionally, the Brood woman appearing after the Dragon Zombie has been slayed states that the zombie in question wasn't a dragon at all, despite being a bag of bones shared every thinkable trait with your average Brood. This would then mean that he was in fact Endless.
Anyway, this is my take on the timeline. Feel free to post your opinions and own timelines!