Cfrgtkky

In real life, fungi are not plants; they are their own separate kingdom (I'm no biologist, so here's a quote from Wikipedia, for what it's worth):

These organisms are classified as a kingdom, fungi, which is separate from the other eukaryotic life kingdoms of plants and animals.

Now on to D&D 5e...

Speak with plants (PHB, p. 277) says:

You imbue plants within 30 feet of you with limited sentience and animation ...

[...]

If a plant creature is in the area, you can communicate with it as if you shared a common language, but you gain no magical ability to influence it.

The spell repeatedly mentions plants, and also makes a mention of plant creatures at the end.

The creatures Myconids (MM, p. 230) and Fungi (MM, p. 137) are both listed as plant type creatures under their stat blocks, and yet they are clearly fungi rather than plants from a narrative perspective. Here's a D&DBeyond search that lists all of these creatures.

Does this mean that speak with plants would work on a mushroom? Not a "creature" (since that's made explicit by the fact that it says "plant creature" and Fungi/Myconids are listed as plant type creatures), just a mundane mushroom, such as some Barrelstalk (Out of the Abyss, p. 22) and whatever else is listed under that section of OotA.

The reason I ask is because I'm running OotA and one of my players is currently playing a deep gnome ranger who is very enthusiastic about mushrooms, and we hit level 11, at which point rangers can pick another 3rd level spell, and the player spotted speak with plants and wanted to use it to talk to mushrooms.

Given that we're in the Underdark and will be staying down here for the rest of the adventure and then some, we're not going to be seeing any plantlife for a long time, so I've ruled that it does work on mundane mushrooms as it would on mundane plants, since otherwise the spell is useless down in the Underdark, but I just wanted to see if this is RAW (or at least RAI) as well.

Yes, the D&D world identifies mushrooms as plants

Historically, the word "plant" referred to both green plants and mushrooms. The microscopic fungi responsible for certain diseases wouldn't have occurred to people as something that could exist until the germ theory of disease gained prominence (late 19th century). D&D means "plant" in this older sense.

This can be hard to stomach. I enjoy D&D in part for the mechanics; I like picking apart the details of how things work and figuring out whether a given mechanic applies in a given situation.

Within the last several decades, the very intelligent people who make careers out of studying plants and fungi reached an overwhelming consensus that fungi are actually more closely related to animals than to plants. They obsessed over the finer details of how things work, and they figured out that plants and fungi are governed by wholly different sets of mechanics, to the point where it's just wrong to call a fungus a plant or vice versa.

Back up, though. Spells like "Speak with Plants" don't say "members of the plant kingdom". They simply refer to "plants". The Player's Handbook is a non-scientific document. It never says what definition of "plant" to use. Several things about the 5e rules point to an older, less-informed understanding of plants.

Myconids

Consult the Monster Manual's definitions of creature types.

Plants in this context are vegetable creatures, not ordinary flora.
Most of them are ambulatory, and some are carnivorous. The
quintessential plants are the shambling mound and the treant. Fungal
creatures such as the gas spore and the myconid also fall into this
category (Monster Manual p. 7).

I'll point out that, by a scientific definition, none of the monsters listed are plants or fungi. As we understand them in the real world, plants and fungi lack muscles and central nervous systems. A scientist would see a shambling mound or myconid and wonder, "How does it move? How does it make decisions?" Many research projects later, the scientific community would revise its definitions (with the details depending on what research revealed about how these creatures functioned).

There are two takeaways here. First, you're just looking for trouble if you try to get too scientific about what words mean in D&D. Second, this is an example where the rules explicitly don't make a distinction between plants and fungus.

This doesn't fully answer your question, though. Speak with Plants targets ordinary flora, whereas the quoted passage explicitly says it's not talking about ordinary flora.

We don't have the right word

Pretend I'm right-- "Plants" in the spell Speak with Plants refers to the non-locomotive, macroscopic organisms attached to the surfaces of wilderness areas (as well as some other places). What's the correct catch-all word for that? If you limit yourself to scientifically-precise words, I don't think we have one. I think this is a case where, instead of taking half a page to write out what does or doesn't count as a plant, the rules-writers figured it would be less clunky and more fun to use the vague term and let the DM exercise discretion.

How In-Game Characters Would See It

D&D takes place in a bygone technological era, back before people distinguished between plants and mushrooms.

You arm your troops with swords, pole arms, and bows. Full plate armor is still the best defense available. There are no steam engines; coal is used to heat stuff, not to make machine parts move. Assume a parallel level of understanding of mycology. From the Wikipedia page:

Historically, mycology was a branch of botany because, although fungi
are evolutionarily more closely related to animals than to plants,
this was not recognized until a few decades ago.

...

The Greek philosopher Theophrastos of Eresos (371-288 B.C.) was perhaps the first to try to systematically classify plants; mushrooms were considered to be plants missing certain organs.

...

The Middle Ages saw little advancement in the body of knowledge about fungi. Rather, the invention of the printing press allowed some authors to disseminate superstitions and misconceptions about the fungi that had been perpetuated by the classical authors.

The start of the modern age of mycology begins with Pier Antonio
Micheli's 1737 publication of Nova plantarum genera.

Sure, you could declare that yours is a world where people actually do understand the difference between plants and mushrooms. By default, though, I think it's understood that characters in D&D have a pre-industrial-revolution understanding of science.

How Gary Gygax et al. saw it

In a similar vein to the previous point, it's unclear to me whether the "Mushrooms aren't plants!" idea would have reached the writers of the first-generation AD&D source books. Sure, 5e isn't 1e, but 5e does draw its inspiration and terminology from previous editions. Calling a fungus a plant feels like a linguistic throwback nowadays, but it's not like the 5e writers invented the throwback from nothing; they borrowed it from an era when it wasn't so much of a throwback.

What's the point of a druid?

Druids love nature. Spells like Plant Growth and Speak with Plants give druids a tangible benefit when they're in wilderness or agricultural areas.

The ecological details of The Underdark are different: Instead of green plants, The Underdark has mushrooms. Despite this, The Underdark still has wilderness and agricultural areas. Does a druid's magic recognize a field of mushrooms as being relevantly similar to a field of barley? I believe the answer is "yes".

The Player's Handbook explicitly says that a Circle of the Land druid can be initiated in The Underdark (PHB p. 68). The Underdark has druids; druids should function properly in The Underdark.

Yes, mushrooms are plants for the purposes of D&D classification.

Ok, I've said "yes" rather authoritatively, but the D&D books don't really go into that much detail on biological classification. Scientific biological analysis doesn't really apply to a fantasy world of elementals, constructs and angels!

However, given that there isn't a specific monster type of "fungi", and given that in DnD Beyond, if you filter by "Plant" you also get creatures such as the shrieker and violet fungus, it is reasonable to say that mushrooms count as the plant type in this context.

Yes, mushrooms are plants in this context.

The Myconids make this rather clear. If "mushroom man" can be classified as "plant creature," then the mushroom part must be what lets it count as a plant. As PJRZ mentioned, Violet Fungus and Shriekers, both 'normal' mushrooms, also count as Plants. In D&D, there is no real need to make a distinction.

The plain English reading also would seem to support this. "I've got lots of wild plants growing in my backyard: ferns, wildflowers, bushes, mushrooms, and even a small sapling!" Only a biologist (or an eager biology student) would object to mushrooms being included in that description.