The classic example (which is probably an urban myth) is a foreign learner saying "I ate a chicken last night" when he meant "I ate some chicken last night".
Mistaking chicken (the meat) for a countable noun makes it sound like he ate an entire bird.
english.stackexchange.com/users/None/
I think you are correct in that if a noun is unambiguously mass or count then the qualifier doesn't matter, the meaning can be taken from the noun.
However compare
: I ate too much fish vs.
I ate too many fish.
If the noun is ambiguous in that way, then the meaning must be taken from the qualifier.
So if you mistook "fish" for a term that was always mass or common, then you could produce an ambiguous statement.
(Inspired by Shinto's example)
english.stackexchange.com/users/122/cindi
Some words have different meanings depending on count/ noncount usage.
If you turn in "a paper", it means you have given the teacher an assignment, while simply having "some paper" means you have plain paper with nothing on it.
I have two papers, means I have two assignments.
I have two pieces of paper means I have two pages.
I have
a chicken means I have a bird, and I have some chicken means I have some bird meat.
Most words only have one meaning and usage, so confusing the two just sounds awkward rather than truly confusing like the paper and chicken examples.
english.stackexchange.com/users/914/Picturepocket
I was recently in a grocery store where the express lane was marked "10 Items or Fewer."
I made a mental note that that was a classy chain, so in a marketing sense, getting it right made a difference.
english.stackexchange.com/users/67/moioci
You can't enroll in Tattooing 101 because you have too many classes.
You can't enroll in Tattooing 101 because you have too much class.
You cannot make too many glasses if you don't have much glass.
You can make fewer glasses if you have less glass.
You can only make a little glass if you only have a little glass.
My uncle has too much wood in his cabinet shop; there's not enough space left for a new planer.
My uncle has too many woods in his cabinet shop; there's no need for him to stock zebra wood.
My sister doesn't like that radio station; she says it has too much blues.
My sister doesn't like that wallpaper; she says it has too much blue.
My sister doesn't like that wallpaper; she says it has too many blues.
english.stackexchange.com/users/3800/oosterwal
Here's a real life example.
"Here's a box full of books, most of them are quite good.
Here's another box.
It has less good books in."
"Here's a box full of books, most of them are quite good.
Here's another box.
It has fewer good books in.
" (1) feels like it should be parsed as follows: "It has [less good] books in".
That is, the books in the box are of (uniformly?)
lower quality.
"less good" is behaving like a compound adjective.(2) on the other hand, seems to suggest "It has fewer [good books] in".
"Good books" is behaving like a compound noun.
The suggestion this time is that the books are possibly of similar quality, it's just that the box isn't as full: there are fewer books.
english.stackexchange.com/users/879/Seamus
Less/fewer, too much/too many, amount/number... When people get these things wrong, it bugs me. But I cannot think of a situation where mistaking a mass noun for a count noun (or vice versa) would ever introduce any kind of ambiguity.  Is there such a case?
Mass nouns and counts nouns. Does getting it wrong ever matter?
english.stackexchange.com
<grammaticality><comparatives><uncountable-nouns>
english.stackexchange.com/questions/2180
english.stackexchange.com/users/879/Seamus
If the noun is unambiguously mass or count, the qualifier probably doesn't matter. Whilst there are situations where mistakenly using a countable noun instead of a mass noun would lead to confusion, in general, there are very few. For example, mistaking chicken (the meat) for a countable noun makes it sound like someone ate an entire bird. One example (of many) would be "I ate a chicken last night" instead of "I ate some chicken last night" - the non-countable is treated as if it is countable thereby creating confusion of meaning.
If the noun is unambiguously mass or count, the qualifier probably doesn't matter. Whilst there are situations where mistakenly using a countable noun instead of a mass noun would lead to confusion, in general, there are very few. For example, mistaking chicken (the meat) for a countable noun makes it sound like someone ate an entire bird. Some words have different meanings depending on count/ noncount usage.  If you turn in "a paper", it means you have given the teacher an assignment, while simply having "some paper" means you have plain paper with nothing on it.  Most words only have one meaning and usage, so confusing the two just sounds awkward rather than truly confusing like the paper and chicken examples.     
If a noun is unambiguously a mass or a count then the qualifier doesn't matter - the meaning can be taken from the noun.  Some words have different meanings depending on count/noncount usage.   Some words have a different meaning depending on whether they are treated as mass nouns or countable nouns. One example is the word "chicken".
Mistaking chicken (the meat) for a countable noun (A BIRD) makes it sound like an entire bird. The potential for confusion arises when a noun is not unanbiguously mass or count, e.g. chicekn. One example is paper (can be either). Another example is "glass" (not countable) versus" glasses" (countable) - so "You cannot make too many glasses if you don't have much glass".
Some words have different meanings depending on count/ noncount usage.  If you turn in "a paper", it means you have given the teacher an assignment, while simply having "some paper" means you have plain paper with nothing on it.   Most words only have one meaning and usage, so confusing the two just sounds awkward rather than truly confusing like the paper and chicken examples.     
The word "chicken" is one word where the meaning of a sentence will change depending on whether it is countable or not. Some words have a different meaning depending on whether they are countable nouns or not.