I have found the textual evidence [for the reading "If you ask me anything in my name..." at John 14:14] sufficiently strong to write to the WTS about this verse and will share the reply when it comes.
The WTS responded as follows :
We thank you for your letter in which you draw to our attention the translation of John 14:14 according to the Kingdom Interlinear Translation of the Greek Scriptures. You wonder why the word "me" of the literal word-for-word translation is omitted in the regular rendering of the New World Translation.
The reason for the difference relates to the different Greek versions. Although the Westcott and Hort text, used as the basis for the Kingdom Interlinear, includes the word "me," there are other Greek versions which omit the word. Among these is the Greek text used as the basis of The Emphatic Diaglott, which reads: "If anything you may ask in the name of me, I will do." The Rotherham translation reads: "If anything you shall ask [me] in my name the same I will do." By enclosing the "me" in brackets, Rotherham explains that the word is supplied, suggesting that there is some disagreement as to whether or not it should actually appear in the Greek text. Our prayers, of course, ascend to Jehovah through Jesus Christ. He had already told his disciples in the preceding verse that "whatever it is that you ask in my name, I will do this, in order that the Father may be glorified in connection with the Son." Since the ascension of Jesus to heaven and the outpouring of the holy spirit at Pentecost 33 C.E., holy spirit has been given to individuals on earth through Jesus Christ and, in turn, our prayers ascend to the Father through Jesus Christ. So, while our petitions are just to the Father, we are, in effect, asking Jesus Christ to help us with our spiritual and material needs. So whether a translator wishes to put in the extra "me" or not, the meaning is the same. - Please see also John 15:16; 16:23.
A similar view is taken by many other translators. For example, the footnote, noting the authorities that read "me," is added to the New English Bible, Revised Standard Version, Today's English Version, the Revised Authorised and Weymouth. But they still leave it out of the main text as does the New World Translation. Others that leave "me" out without any explanation include Young's, The Twentieth Century New Testament, Fenton, Darby, Schonfield, The Jerusalem Bible, Barclay and the careful translation by C. B. Williams. Interestingly, although Alford included it in the textual apparatus of his Greek Testament, and showed that the authorities rejecting it were 'relatively late,' yet in the New Testament translation published in 1869, he did not consider it necessary to amend the Authorised Version rendering. So the vital question of context must be considered in addition to all other factors and in this case a large number of translations have taken that as decisive. In Moffatt's translation, which includes "me," this explanation is offered in the Moffatt New Testament Commentary on John, by G. H. C. MacGregor, page 308: "It seems redundant with 'in my name,' and moreover, the only prayer of which this gospel speaks is prayer to the Father in the name and spirit of Christ not prayer to Christ direct."
We trust the above information proves helpful to you and explains why the New World Translation is justified in rendering John 14:14 as it does. We take this opportunity to send you our warm Christian love etc.
In the translations that I have at home "me" is included in the main text in the Revised Version, in George Lamsa's New Testament According to the Eastern Text, and in the Confraternity Version. In the New English Bible it is not in the main text but the footnote reads "Some witnesses insert me".
My conclusion is that whether or not the decision to drop "me" was theological rather than textual, the fact that neither the New English Bible nor the Jerusalem Bible included it in the body of the text gives a lot of weight to their decision. Neither the inclusion or exclusion of "me" in this text should be described as a translation error but may be regarded as a theological predisposition.
Earnest