In "Forsaken," Dax posits that non-biological life could exist---mechanical life, for instance. We know this is the case since nanites, among other mechanical life, exist. My question is whether Dax (or any knowledgeable 24th century person) would consider energy beings such as Douwd, Q, and Organians to be "biological," or if "biological" only refers to organic life forms: Hirogen, Andorians, Allasomorphs, Excalbians, Changelings, etc.
It would likely depend on the context of the discussion. A conversation about a viral plague would no doubt make a distinction between "biological" species (i.e. those who could contract the virus) and non-biological species (i.e. mechanical, energetic, or any other type which would be immune).
Whereas, a discussion of organic vs synthetic life (such as takes center-stage in the Mass Effect universe) would probably include any species that used to be biological but ultimately evolved into something else (such as many speculate the Q to have done) under the heading "biological," as opposed to purely artificial lifeforms (such as Data). In such a case, the "biological" label becomes less about their actual current reality and more about their history or cultural/political alignment, in opposition to the vastly different "artificial" category of species.
Of course, in a straight, technical discussion, beings without biology (such as Douwd or Q) are not biological. But given the wild-and-wacky world of Star Trek, there are plenty of circumstances where it might be expedient to gloss over the distinctions and group them all under the heading "biological" for convenience.
So technically, no. But practically, maybe.
Biology is the study of life; so if you consider that the non-organic entities such as synthetic life or Q are living beings then yes, you must consider them as biological.
In one episode of Voyager (I don't remember which one exactly), Q as a child (with Q2?); therefore I don't see how it would be possible to not consider him as a living being and therefore, subject to the study of life.
For some synthetical lifes such as the nanites; I suppose that the final answer depends on your consideration of them as being living or not.
By definition, "biological" means "living". Are energy beings living? It may depend upon your definition of life. IMHO, the defining properties of life reduce to the capacity for growth, reproduction, and evolution. Therefore, for an energy being to be biological, it must be able to grow, reproduce, and evolve.
Just because a being is non-corporeal doesn't mean it isn't biological. The energy being in ST:TOS "Obsession" needed to feed, and was thought to be preparing to spawn. Since it had to have come into existence somehow, it's reasonable to conclude it must have evolved from something else. It seems to satisfy all the criteria of something living, so it could be considered biological.
The Q are believed if not self-declared to have evolved from beings not all that unlike humans (which is why the Q is interested in humans - seeing potential future rivals to themselves). The Q, despite their nature contemporaneous with TNG,DS9,Voyager and especially given "Junior" (Voyager "Q2") would seem to qualify as biological, although it is said that they haven't reproduced in a very long time. Given that Trelane had parents, his race must also be biological.
The Organians might not be biological because nothing is ever said about them growing and reproducing; if they have ceased to reproduce, they may have become eternal/immortal and reached a sort of "end of the road"; they would therefore no longer qualify as biological.