This may not be what you are looking for, because it is more a matter of language use and culture than have language structure. Nevertheless it corresponds to the phenomenon described. In posh viennese coffeehouses like Demel you might be addressed in the following old fashioned way by your server: "Mögen der Herr noch etwas Sauce?"

In Büchner's Woyzeck the captain says to Woyzeck. "Langsam, Woyzeck, langsam; eins nach dem andern! Er macht mir ganz schwindlig. [...] Er sieht immer so verhetzt aus! Ein guter Mensch tut das nicht, ein guter Mensch, der sein gutes Gewissen hat. – Red er doch was Woyzeck! Was ist heut für Wetter?" The use of the 3rd person as a factual 2nd person has mainly a social significance.

Turkish language regularly omits pronoun, and has person agreement as the verb is agglutinative in form. However, pronoun omission does not serve to reflect politeness. It is often simply redundant. When pronoun is used, it is often to emphasise the person. Not sure if this helps.

What March says of Turkish holds for Slavic languages like Polish or Russian.  However, Russians say that the use of impersonal requests (with mne 'I, Dat' and no verb agreement), is more polite than personal requests (with ja 'I, Nom' and 1st person agreement).

In Polish language every personal propoun, whether singular or plural pronoun, is omitted very often, maybe even almost always, regardless of the register (sociolinguitics) that is formal, semiformal or informal . Simply speaking, the number of inflectional and derivational suffixes seems overhelming so that such omission does not have a negative impact on the contextual meaning of the uttery. In translation, especially Machine Translation all these language pair phenomena between Polish and the target languages constitute an substantial obstacle. Perhap French could be also found a synthetic language, however, using personal pronoun is obligatory for following the correct grammatical rules. I guess, other synthetic languages could be German, Russian, Czech and definitely Finnish.

Hello Johnaton,

Russian is the case, though pronouns there are not usually omitted. More often in spoken language, almost never in writing. The same for Ukrainian. Here, the omission is rather due to economy, than politeness.

Spanish might interest you more. In Spanish it is almost a rule to omit personal pronouns both in speaking and writing (my teacher often emphasized that).

But I do not think Russians (that has verb agreement with grammatical subject) omit subject in order to be polite. I think it's usual in oral speech, and I am not sure it is directly related to politeness.

My phrasing led Rudnitskaya to a slight misunderstanding of what I intended to say:  mne xocetsja, to an older generation of Russians, is more polite than ja xocu.  One old teacher used to say, Tsar' xocet.  Cto vam xocetsja?  'The tsar wants (something).  What is wanted for you?'  The form considered, at least by his generation, as polite, has a subject and verb agreement, but it isn't the first person subject with verb agreement, where the desired object would be in the accusative case.

In spoken Polish, the subject pronoun is often omitted (as redundant?), because verb agreement tells all.  Russian generally does not omit subject pronoun in past tense, because the verb shows only gender agreement, not person.  Kudriavtseva is right about Ukrainian, though I simply can't say anything about politeness in this case.

Thanks to all who've answered so far. I realise now that my question wasn't entirely clear (I did have more detail but had to omit some parts to meet the character limit). I should clarify that by 'pronoun avoidance' I don't just mean pronoun omission, or 'pro drop' (e.g., "Went to the beach today."). Rather, I mean usages where the speaker or hearer is referred to by a name or kin term rather than with a pronoun (e.g., "Would sir like another drink?"; or, a mother speaking to her child: "Mummy has to go now."). Please note that this should also be distinguished from vocative uses (e.g., "Sir, would you like another drink?") which I am not interested in here. I am also not really interested here in politeness per se, it's just that politeness is just a common (but not the only) motivation for avoiding pronouns in some languages, and I hoped this would help others identify the kinds of examples I'm interested in.

Specifically, I'd like to find examples where the use of proper names/titles/kin terms are used in lieu of 1st/2nd person subject pronouns (not as vocative terms of address, etc.) in languages that have person agreement, and I'm interested to see whether the verb takes 3rd person agreement in these contexts (as in the English "Has sir had enough?") or whether it ever takes 1st/2nd person agreement on account of the fact that the subject (though formally 3rd person) refers to the speaker or hearer.

Some of the responses here are very interesting, though I need a bit more information in some cases. David, in your examples, it looks like titles/nouns are being used instead of pronouns, but I can't tell from the English translations whether the verb agreement is 3rd person, or whether it is perhaps 2nd person and 1st person respectively in the original Hebrew text. I can't read Hebrew, so I'd appreciate it if you could help me out here!

Polish fits here.  Pan się pomylił 'Sir has made a mistake' is quite formal.  Pomyliłeś *you've made a mistake* is informal, even familiar.  Pan się pomyliłeś *?no E equivalent* is in between:  Pan 'sir' is used as the name, but instead of third-person agreement, as in the first example, there is second-perrson singular, as in the second example.  So the morphology and lexis, we have an in-between sentence with an in-between meaning.

Russian also fits. You can say "Gospodin skazal" (Sir has said)  only with 3rd person agreement, not 2 nd. If the person is very distinguished you can use 3rd person plural agreement: "Ego velichestvo otkushali" (His Majesty have had a meal). All of these forms are old-fashioned and now used when you want to make fun of your relatives and friends.

I don't know if this is particularly to the topic, but a number of years ago, Robert Dole was running for President (or vice-President, something of that sort, who remembers?) and he was fond of using his name to replace the first singular pronoun.  For instance, (this is a made-up sentence, but it fits the mold), "Bob Dole wants you to vote.....", and he did it more and more as the campaigned continued, which in the States is far too long, so people began to invent similar sentences or phrases using another well-known person.  The funniest one was when someone took the title of the popular television series "I, Claudius" and turned it into "Claudius, Claudius".

Polish is a pro-drop language with person, number and gender agreement, so pronouns are usually omitted, because they are perfectly dispensable. Politeness has nothing to do here.

Polish does not have vocative uses as described above of the type: "Sir, would you like another drink?". These meanings are always expressed with 3rd person singular: "Czy zechciałby Pan... Czy zechciałaby Pani...?" or plural: "Czy zechcieliby Państwo...?", meaning "Would you like...?" in each case.

The second person singular form provided by William Sullivan is most likely regional and quite odd to me. I may have heard it once or twice in my life, being from another part from Poland, but the 2nd person form is perfectly acceptable to me in the plural: "Czy zechcielibyście Państwo...?".

While Pan, Pani, Państwo, Panie, Panowie, etc. is usually used to address strangers or superiors, in the older generation of Poles the 3rd person form was used to show respect while addressing parents, like in: "Czy mama poszła...?" (Did Mother go...?).

Jonathon, your actual question is different to what I had understood. Again, returning to the Turkish example... The replacement of pronoun with the title is possible for two reasons I can think of. 1. Politeness: waiter addressing sir "would mister like any water?" / Bay su ister miydi?. The personal ending for 2nd person formal takes third person agreement in this case. Note that in Turkish, the word mister would be the first word in the sentence whether it replaces the pronoun or it is vocative. In "Bay, su ister miydiniz?"  / Mister, would you like water?, there is a comma and the verb ending has second person agreement. In spoken language, the comma is not evident. It is the pronominal suffix of the verb that indicates whether the word sir has been utilised to replace the pronoun or served a vocative role. So, the pronominal ending plays a large role to distinguish the two uses.

2. Formalising first person/self to third person. Using your example, mother stating to child "Mummy has to go now" / Anne simdi gitmek zorunda (child less likely to reduce). Another example, "Mummy loves you very much" / Anne seni cok seviyor (serves to make the statement more objective?). In both cases even though the speaker addresses the self, by virtue of changing the subject, the verb ending takes third person agreement. Note that the use of mummy puts a distance to speaker, in the first example, serving authority and in the second, warmth (not dissimilar to how it would be interpreted in English).  

Thanks everyone, there are some helpful examples here. :)

Jonathon:

This can be applied to the Arabic language. Arabic is highly for person, number and gender. However, in highly formal situation, in order to show respect, the second singular person pronoun is changed to plural like in the question, matha tuhib-a " what like-you.sg?" is changed into matha tuhib-uun? "what like-you.pl?". Similarly, if you talk about a president, for example, the third person pronoun huwa 'he' is changed into fakhamatahu 'his excellency'. 

I think there are two aspects of the question: a) avoiding pronouns and b) avoiding agents for subjects. There is, perhaps, a relation to politeness, but maybe not so direct one.

In Czech, which (same as Polish or Russian) has the aggreement (congruence) of the subject person and the verb ending, using or not using personal pronouns is more of a question of the information structure (topic-focus articulation, theme-rheme etc.). When the focus is on the subject, it is usualy verbalized (be it in a form of a pronoun or a noun); if the subject is out of focus, then the subject is expressed only by the verb ending (that's what transformationalists call "pro-drop"). I'd say that this has little to do with politeness.

What does relate to politeness is the strategy of deagentivization - not making 2nd person the agent/subject of the verb is polite towards the hearer, because one of the semantic propperties of the prototypical subject is the responsibility (that goes hand in hand with the agentivity) and stripping off the responsibility is polite. There are several ways to deagentivize. One is to use passive construction, others is reflexive passives (with "se" and 3rd person sg. verb ending, similar to the "pasiva refleja" in Spanish) and there are more. So instead of "Mohl byste...?" ("Could you...?"), we can say "Bylo by možno...?" ("Would it be possible to...?") (note that both cases are without personal pronouns in Czech, but the second one is more polite, because the hearer is not the subject).

It is clear, that using or not using a personal pronoun is not the question of politeness here. The question is using it in the subject position. In the example below, the pronoun must be used for reasons of the information structure (focalization), but the example ii) is more polite, since the 2nd person pronoun is not in the subject position:

i) "Já to nedokážu. Mohl byste to udělat vy(FOC)?" (I can't do it. Could you(FOC) do it?)

ii) "Já to nedokážu. Bylo by to pro vás(FOC) uskutečnitelné?" (I can't do it. Would it be possible for you(FOC) to do it?)

At least that's my view at the moment.