This Maavor Yabok is brought by Reb Akiva Eiger (OC 119:1) and the Chida in Birkei Yosef (YD 240:4). The Minchas Chinuch also brings the Maavar Yabok in Mitzva 297. He asks, how did Elozor refer to Moshe by name (Bamidbar 31:21, ויאמר אלעזר הכהן אל אנשי הצבא הבאים למלחמה: זאת חקת התורה אשר צוה יהוה את משה) He was his Rebbi, and it's assur to refer to a Rebbi by his name alone. He answers that because Elazar was saying that Hashem had commanded this to Moshe, it would have been wrong to use an honorific, as the Maavar Yabok says.
It has been suggested (e.g., Reb Reuven Margolios in his notes on Sefer Chasidim) that this only applies to tefillos for a sick person, because of the condition of strict judgment- Middas Hadin- but not to general brachos or other types of tefillos. In general tefilla, such as in Bentching, you may say "Hu yevariech es Avi Mori." This is supported by the Ramban's mention of the minhag in Teiman; After the Rambam interceded on behalf of the Jewish community, they inserted into Kaddish the words בחייכון וביומיכון ובחיי דרבנא משה בן מימון (as brought down by Chavel in his Kisvei HaRamban Vol I p. 341). I don't find this convincing, because although the Sefer Chasidim said "refu'ah," his examples are from cases that have nothing to do with refu'ah. I think that the Ramban and the Minhag of Teiman simply didn't hold like Reb Yehuda Hachasid's idea. But Reb Shlomo Zalman Auerbach did say (brought in Or Ephraim 558) that one may use the titles Rabbeinu and Maranan when you call someone for an aliyah and in Yekum Purkan because this emphasizes that Torah she'biksav is incomplete without the mesora of Torah she'baal peh which resides in and is personified by these people.
After the discussion in the comments, and some further thought, there are some additional things that need addressing.
1. The problem with referring to people with titles of honor in Tefilla is dual. First, as the Maavar Yabok says, it is a chutzpa on the part of the mispallel to honor a human while talking to the Ribono shel Olam. This focuses on the problem for the mispallel, that the mispallel is like a מורד במלכות, but it has nothing to do with the effect on the subject of the tefilla. Second, (and this is somewhat speculative,) it is not good for the subject of the tefilla. It could be that אין גבהות לפני המקום teaches that using titles in tefilla offends the Ribono shel Olam and nullifies any Ritzui you are looking for. It makes the tefilla worthless. You ask for rachamim, and you offend the Ribono shel Olam by showing honor to one of His little creations- such a tefilla will not yield any benefit at all, and maybe it is מעורר קטיגוריא למעלה chas veshalom. So the first comment, that said that my baal tefilla has a petur of oneis, he's right as far as the baal tefilla, but it won't help the subject of the tefilla. The second comment, that mentioned the k'peida of some gedolim, highlights this second issue.
2. I don't believe the minhag is like the Sefer Chasidim/Maavar Yabok/Birkei Yosef/Reb Akiva Eiger. Maasim bechol yom is proof enough. So what do we do with all these poskim? Maybe we will apply them only during Shmone Esrei, when you are mamosh "עומד לפני המלך". But stam during Tefilla, or when you're calling someone up for an aliyah, the problem is not as serious, and in those cases I don't believe we're noheig like that. Honoring someone that you're mechuyav to honor, either because of kavod hatorah or kibbud av or hakaras hatov, that is not an affront to the Ribono shel Olam. When you're standing in front of the Ribono shel Olam, then you have to be extremely cautious, but otherwise, its not a problem.
Alternatively, we can say that the Ramban that brings the minhag of Teiman to say ובחיי דרבנא משה in kaddish contradicts these shittos, and we can do like the Ramban.
3. Even if you want to say we do hold like the osrim, I would say that we should limit that to tefilla for Refu'ah. When you're asking for refu'ah, you are focusing on human helplessness and our dependence for our very breath on the chesed of the Ribono shel Olam. For such a tefilla, assertions of chashivus are truly incongruous and counterproductive. But in other cases, I don't think our minhag is to worry about that.
So, in short, the additional points are:
1. Those that prohibit honorifics in tefilla do so for two reasons: it is a sin for the person saying the tefilla, and it is not healthy for the person for whom you're davening.
2. I believe our minhag is to not worry about this prohibition, or, if we do, we only are makpid in Shmoneh Esrei.
3. Alternatively, maybe it's only something to worry about in tefillos for refuah for a choleh, but not when you call up for aliyos or other tefillos.