The following is a list of passages that were added to Scripture or modified from the original in some way. This page is not calling any part of Scripture itself false, but rather, it's the modification of Scripture that is being called false. For passages who's English translations are wrong, see Mistranslated Scriptures.
There aren't very many instances of false Scriptures in the Tanakh.
"And the men turned their faces from thence, and went toward Sodom: but Abraham stood yet before the Lord [KJV]." In the original text, it's the Lord who is standing before Abraham. For some reason, the scribes in charge of copying Torah didn't like the idea of the Creator standing before his creation, so they reversed the roles.
"The Spirit of the Lord God is upon me; because the Lord hath anointed me to preach good tidings unto the meek; he hath sent me to bind up the brokenhearted, to proclaim liberty to the captives, and the opening of the prison to them that are bound; to proclaim the acceptable year of the Lord... [KJV]." A statement about the blind receiving sight was removed from this passage of Scripture by the Pharisees after Jesus' ministry on Earth. In Luke 4:18-19, Jesus quotes the passage as saying "The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives, and recovering of sight to the blind, to set at liberty them that are bruised, to preach the acceptable year of the Lord." The reason for doing this is quite simple: They wanted to make Jesus look like a false prophet in order to justify their hatred of him. And what better way to do that than to make it look like he misquoted one of the prophets? Typically, when someone modifies Scripture, they add to it. This time, they have taken away from it.
The vast majority of false Scriptures come from the "New Testament". The Church, becoming more and more corrupt after the death of the apostles, added to and modified the New Testament texts in various places in order to be able to claim that Scripture supports their unbiblical teachings. Note that this section is incomplete.
"Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: [KJV]." This passage of Scripture was modified to include baptism and a reference to the Trinity very early on, either in the process of translating the manuscript from Hebrew to Greek, or when the number of Greek manuscripts were few and far between. We know from Acts onward that not all of the disciples baptized those whom accepted the Gospel message. And when they did, it was always in the name of Jesus, not in the name of the Trinity. Rather, they were teaching all nations in the name of Jesus. This apparent act of disobedience was never corrected either by prophet, by vision, or by a fellow believer. This proves that the command, as it is presented to us, is not the command was given. It was most likely the case that the original command was "Go ye therefore, and teach all nations in my name." This matches what we see recorded in the book of Acts and explains why absolutely no correction was given: There was no need for correction because there was no disobedience.