Some find it difficult to form an "r" sound, and some are able to, but just don't. I'm looking for a word which means "not pronouncing r's", without implying inability to pronounce them, though that may be the case.
Specifically, I'd like a word which mean "not pronouncing any r's", but if there's a word which means "not pronouncing some r's" (as in many dialects), that would be interesting to know too.
The Wikipedia article on rhotacism gives one definition of that word as “the inability to or difficulty in pronouncing r”.
Non-rhotic may be what you're looking for:
English pronunciation can be divided into two main accent groups: a rhotic (pronounced /ˈroʊtɨk/, sometimes /ˈrɒtɨk/) speaker pronounces a rhotic consonant in words like hard; a non-rhotic speaker does not. That is, rhotic speakers pronounce /r/ in all positions, while non-rhotic speakers pronounce /r/ only if it is followed by a vowel sound in the same phrase or prosodic unit (see "linking and intrusive R").
In linguistic terms, non-rhotic accents are said to exclude the sound [r] from the syllable coda before a consonant or prosodic break. This is commonly (if misleadingly) referred to as "post-vocalic R".
I know this question has been answered already, but at least one didn't nail your question I don't feel. Rhotacism is specifically pronouncing an R in circumstances where you wouldn't otherwise. In other words it's the pronunciation of the R when certain letter and phonemic combinations are found. Also, it's not the standard English R pronunciation, it is the alveolar tap or trill, found in languages like Italian, Spanish, etc. (Also found in some English, but not standard). These changes in English where we might make an alveolar tap or trill appears in informal things like:
"I got to get it."
"I goda gedit"
^ ^ // Pointers point to alveolar taps or rarely trills
Rhotacism is also the impairment of not being able to make this sound, notice that the Wikipedia article says that the people who have this condition are usually the:
Finnish, Hungarian, Italian, Indonesian, Bulgarian, Romanian, Russian and Spanish
Rhotacism (impairment)
Which shows that it's not this sound that English speakers make when they pronounce Rs in words.
The real phenomenon, of NOT pronouncing the R at least in English is:
non-rhoticity
non-rhotic was given in the other answer, but what the heck I've added it here anyway, plus it's also the noun form, which is what you asked for.
Is there a word for "not pronouncing any r's"?
The short answer is “No.”
The long answer is "It is not "r-dropping" because there never was an [r] to drop. It is not “rhotic” or any variation.”
Rhotic adj. (and other forms of the word) do not cover what the OP is asking. Rhotic (et al) was introduced into English in 1968 by J. C. Wells and has a specific meaning – it describes how [r] is pronounced: see below.
OED
rhotic adj.
Of or relating to a variety or dialect of English in which r is pronounced not only in prevocalic position but also before a consonant or word-finally; characterized by the pronunciation of r. Cf. non-rhotic adj.
1968 It was possible to divide respondents into three categories: A. (non-rhotic) Those who had nonprevocalic r-colouring neither for -er nor for -a; B. (rhotic) Those who had nonprevocalic r-colouring for -er but not for -a; C. (hyperrhotic). J. C. Wells in Progress Rep. Phonetics Lab. Univ. Coll. London (unpublished) June 56
non-rhotic adj.
Of or relating to a dialect in which r is only pronounced in prevocalic position; characterized by such a pronunciation.
1968 It was possible to divide respondents into three categories: A. (non-rhotic) Those who had nonprevocalic r-colouring neither for -er nor for -a, [etc.]. J. C. Wells in Progress Rep. Phonetics Lab. Univ. Coll. London (unpublished) June 56
Some languages, notably Japanese (and, I believe, Chinese) do not have the English [r], be it rhotic or non-rhotic. The nearest sound is somewhere between [l] and [r].
There is no word that specifically refers to the characteristic of not pronouncing [r] at all.
(There is a medical/physical/genetic condition of the tongue that prevents the pronunciation of [r] but (i) there is no context to suggest that this is what the OP is looking for, (ii) if this is what the OP wants, a medical forum would be better.)
