It's possible the following answer is long on speculation and short on evidence. Remove it if you think so. I'm posting anyway in the hope that it might clear the ground for some further answers which are a bit longer on evidence.
The short answer is that all complex translation involves interpretation, and it's not possible to separate that interpretation from the interests of the translator. But given the Christian church's use(s) of the texts of the bible, it's also pretty hard for obvious, malicious translations to go unquestioned. Except, perhaps when they do.
I make the following assumptions about biblical translation:
Any re-presentation of a text is done with some motivation. A translation is like an edition of a text and is being re-presented to a new audience.
All translation involves interpretation. There is no such thing as a straight translation with "unchanged meaning", especially of a complex set of texts like the Bible.
So any translation of the Bible, and the interpretation that involves will be 'motivated', and therefore connected to the interests of the person or group who undertakes the task.
An example of a place where Martin Luther's translation has been accused of being too influenced by his theology is Romans 3:28, where he translated that "So now we hold, that man is justified without the help of the works of the law, alone through faith" (So halten wyrs nu, das der mensch gerechtfertiget werde, on zu thun der werck des gesetzs, alleyn durch den glawben). The Greek text does not include word that corresponds to 'alone'. Luther justified his German translation on the basis of the sense of the original and the requirements of German idiom, but his version does rather emphasize Luther's own theological concern that salvation comes only 'by faith' over against the contemporary Catholic emphasis that it must be accompanied by works of charity or that the (Roman) Church itself has an essential role in its administration.
Such questions come up all the time in translation, even with texts that are nowhere near as diverse as the biblical texts.
But (with some important exceptions), the Christian church has generally considered the Greek and Hebrew texts to be the authority on the meaning of any translation. The authority of any version of the Bible depended both on the authority of the translating/publishing/authorizing body, but also on its demonstrated correspondence to its source texts. So any 'serious' edition of the Bible which could be shown to have brazenly manipulated translations would likely lose credibility as a translation, at least eventually. Because of this the Christian church had a notable tendency to preserve its texts, and from time to time there were serious (even enormous) efforts to produce critical editions of the best texts available, or to publish parallel versions which compared respected translations such as the Vulgate or the Septuagint with editions in the original languages. For example, the Hexapla of Origen (3rd C) is an example of a critical Hebrew text comparing 6 versions of the Hebrew Bible (a.k.a the Old Testament). After the invention of the printing press, the Complutensian Polyglot (1520) was a comparison of the available Hebrew, Greek and Latin editions of the time, and Erasmus' New Testament (1516) includes a fresh correlation of Greek texts alongside his own new Latin translation. Now of course for about a thousand years, the Latin Vulgate was considered the definitive Bible for much of European Christendom, so the claim of the authority of the original-languages editions will have to be tempered by that reality. Perhaps someone who knows more about the authority of the original language editions in medieval Christian exegetical debates can say more about this.
So while it's not hard to see ideological influence on biblical translation, Christian respect for the original editions is one thing that has worked against deliberate distortion of meaning and the promotion of narrow group interests by published biblical texts. Of course there are still lots of other ways to achieve those ends.