Some vegetarians avoid eating mushrooms. Are mushrooms a non-vegetarian product? if so, what makes them non-vegetarian, and why do vegetarians avoid eating mushrooms?
In biology, organisms are grouped into a set of categories called, for some reason, kingdoms. These are, of course, somewhat artificial, but the need of biologists to categorise is shared by vegetarians and vegans; so much so that we can use the biology concept of kingdoms to say what it is we eat and do not eat.
In general, vegetarians do not eat animals.
In general, vegans do not eat animals or substances derived from them.
Mushrooms are not in the animal kingdom - they are fungi. They can be considered vegetarian and vegan because they are not animal or animal-derived. A related question asks Is yeast-risen bread (or other baked products) vegan?.
Thinking in a more qualitative way about why veg*ns might be OK with eating mushrooms, I suppose we might not feel compassion for mushrooms or other fungi because, for example, they don't express any obvious distress about being chopped up and eaten.
From the other angle, why might some vegetarians not eat mushrooms, aside from personal reasons as suggested by Ed Grimm's answer, I think many people are not aware of what is "officially" vegetarian, and aren't sure whether mushrooms are plants, animals, or, as I've heard people say, "something in between" (from the biological classification perspective, they are neither). A vegetarian might want to follow the rules but may be unsure what the rules are (because there aren't any, really).
I've never heard of mushrooms being considered non-vegan.
That having been said, there are some people who are allergic to fungus spores and/or fungus protein. There are also people who are really disturbed by fungus texture or resilience. And, as all foods, there are some people who are just put off by the thought of them.
There are probably some other reasons as well, but that's the ones I've personally encountered.
Mushrooms (or fungi) are not plants and are not animals either. Feel free to check out some "phylogenetic tree" diagrams to understand more clearly how they differ. The eatable mushroom is a bit like a "fruit", a fruiting body, meaning it's kind of made to be vulnerable and eventually eaten. The function of the mushroom is to release spores. When you take a mushroom from the forest, the mycelium organism remains unharmed, under the ground.
For these reasons I would rule it as vegan-friendly. I would also conclude that fruits and mushrooms are more vegan than vegetables, because fruits are made to be eaten without causing harm, and roots and leaves aren't. However if your wish is to be completely "plant-based", then mushrooms are not a plant food.
There are many different ways of representing the phylogenetic tree and we cannot be sure of all the ramifications, but everyone agrees that fungi are neither plants or animals, even though they might be considered "closer" to animals than to plants.
Phylogenetic tree (source):
Another possible reason some vegetarians/vegans might avoid mushrooms is because of mushroom farming practices. Mushroom soil is generally a manure-rich soil used for growing mushrooms, and one could argue that mushrooms use animal byproducts in their growth, but in reality, if we based all decisions on that ground, there'd be very little for anyone to eat due to the composition of garden soils, and just top soil in general.
Some mushrooms can eat living creatures like insects. Because a vegan wouldn't eat bugs some of them don't eat the mushrooms, because the mushroom can eat bugs. A famous example is the oyster mushroom. I'll provide a link so you can read about it in further detail.
It's for this same reason that some vegan's avoid eating wild type figs, because figs will also eat a specific insect in the fruiting body.