One should perhaps distinguish between:
- a false perception of a phonetic distinction in someone's own speech or in the language in general
- the feeling that a distinction does exist in the "proper" pronunciation but that speakers "corrupt" this pronunciation by merging phonemes
- lack of awareness of certain phonetic phenomena such as vowel reduction, voicing/devoicing and sandhi
Adding to Tristan's observation about the (lack of) distinction between 'v' and 'b' in Spanish, I think one example of a false perception is the lack of a phonetic distinction between the short stressed realizations of the graphemes 'ä' and 'e' in Standard German. See the discussion here. I myself had this experience of talking to educated native speakers who tried to convince me that there was or must have been a phonetic distinction.
In the case of Hebrew, since it has not been spoken as a first language up until end of 19th century, the distinction between the different phenomena above is particularly relevant. I'm not aware of examples of false perception of a phonetic distinction, but there have indeed been cases when a wide-spread pronunciation of Hebrew was reflected upon by the more perceptive observers from within the community who noticed that it started to diverge from how Hebrew "ought to be pronounced" - see examples in Dovid Katz'es "The phonology of Ashkenazic". It is not always clear whether in these cases the critic follows a tradition which genuinely survived the phonetic merger or whether he cautiously adjusts his own pronunciation (how successfully?) despite the merger he and everyone else already had.
An example of a lack of awareness (3.) might be final devoicing and vowel reduction among some Hebrew speakers of Russian background and lack of distinction between /v/ and /b/ among speakers of Spanish background.
See also a related phenomenon when graphics influences pronunciation.