I can't say it never happened, but I can't think of any specific examples. The vast majority of textual variants were unintentional, essentially the scribal equivalent of what we'd call "typos" today. In some cases, it's clear that the scribe didn't even really understand what he was writing, but was just copying the shapes. In that situation, it would be impossible to make a biased or purposeful omission.

There are certainly clear intentional changes among manuscripts, but these are almost exclusively efforts at clarification. For example, a scribe might replace "he" in the original text with "Jesus" in order to make it clearer to the reader who the pronouns are referring to. In some cases, a scribe might try to correct what he thought was an error made by a previous scribe. However, this also would depend on the scribal tradition - for example, among the Jewish Masoretic scribes there developed a distinction between the "qere" (what is read) and the "ketib" (what is written). When the Masoretes added vowel pointings to the Hebrew text, at certain words it was grammatically unclear how it should be pointed, which would affect the meaning of the word, so they maintained the traditional vowel pointing, but added what they considered to be the correct form in the margin, preserving both.

As for a monk or other scribe making a purposeful omission from a Bible to accomplish some sort of agenda... That seems unlikely. Remember that monks, even in the points of Medieval history where monasteries were notoriously lax and secularized, regarded Scripture as the Holy Word of God, and messing with that could easily have severe repercussions on your immortal soul. And even in an era before the printing press, there were enough manuscripts floating around, and enough familiarity with the text of Scripture, that a suspicious passage could be checked against other texts.

In modern textual criticism and Biblical scholarship, the accusation of some sort of bias or effort to sanitize the text is indeed the basis of one of the main criteria for judging authenticity of a passage: it's called the "criterion of embarrassment". For example, in a Gospel text that makes the disciples look bad or stupid, it's more likely to be a true story faithfully preserved, because the disciples are less likely to have invented it and subsequent scribes less likely to have preserved it if it weren't actually true.