Luke 2:14 - Goodwill(ed) Men
Glory to God in the highest, and on earth peace, good will toward men.
This isn't so much a mistranslation as it is an example of the complicated nature of the history of the New Testament. This phrase is spoken by the hosts of heaven appearing to the shepherds outside of Bethlehem during Luke's version of the birth narrative. The issue is over the word "good will": εὐδοκία eudokía. *First, *eudokía might be better translated as "good favor", it's not a quality that someone can have or gain on their own, but one that they obtain from others. So in this case, the favor is coming from God.
The second, and larger issue, is one of grammar. In English, grammar is usually expressed through word order: subject, verb, direct object. In Greek, grammar is usually expressed through changes to the word's pronunciation and spelling. The problem is that the word eudokía appears as εὐδοκίας eudokías in a good deal of the oldest manuscripts - note the little sigma at the end. That little sigma changes it from the nominative case (or roughly the subject of the verb) to the genitive case (or roughly the ownership case represented in English by the word "of", the book *of John). *That little sigma at the end changes the phrase to read:
Glory to God in the highest, and on earth peace toward humans of good-favor.
Now there's evidence for both readings in some of our oldest manuscripts, but it's easier for scholars to assume that the work was originally written with a sigma that occasionally got forgotten than to imagine that someone decided that they'd purposefully place a sigma at the end of the world. Thus, the majority of scholars (but by no means all of them) feel that the actual reading of this famous verse should read something akin to:
Glory to God in the highest, and peace on earth toward humans of his good favor.
Philippians 2:6 - Robbery
Who, being in the form of God, thought it not robbery to be equal with God.
This verse needs a bit of context. Paul is talking about how Christians should follow the example of Christ. Then this verse follows. Take a quick look at Philippians 2 to see the entirely of what seems to be Paul quoting an early Christian hymn or poem.
The problem with this verse is the word "robbery". In English this verse makes it sound like Jesus felt that it would not be something wrong or undeserved to be equal with God. Some Mormons (though not many, admittedly) feel that this verse is talking about the doctrine of eternal progression whereby even Christ had to progress to become like God.
In actuality, the word underlying "robbery", ἁρπαγμός harpagmós, has an interesting history in Greek. Over its history it morphed from being used as the act of theft to being about the object of theft itself. In this way it's remarkably similar to the English word "plunder", which can refer to either the act of plundering or to that which is plundered. (Note: there's no actual relationship between plunder and harpagmós, they just end up working similarly and means similar things).
This is why this verse would be better translated as:
Who, being in the form of God, thought equality with God as something not to be taken.
This changes the meaning substantially; now the entire hymn is about how Jesus, though he is in the form of God, rejects equality with God and instead humbled himself. The rest of the verses talk about this humility, encouraging the Philippians to also be humble in their circumstances before God.
1 Thessalonians 5:22 - All Appearance
Abstain from all appearance of evil.
This one is actually pretty simple. Whereas many Mormon parents use it to say that actions that they think look evil should be avoided, even if they aren't actually evil. But the word εἶδος eídos ("shape, form, appearance") is referring to evil in the sense that anytime evil appears then it should be avoided. Quite simply, Paul is telling his followers that they should leave all kinds of evil alone. He's not saying that the Thessalonians should care about whether or not their actions seem evil to others.
1 Timothy 4:8 - Bodily Exercise
For bodily exercise profiteth little: but godliness is profitable unto all things, having promise of the life that now is, and of that which is to come.
Again, this one doesn't come up often, but I have heard some people use it to justify Mormons' general sweet teeth (sweet tooths?): "bodily exercise profiteth little". It sounds like Pseudo-Paul (because remember, Paul didn't write either letters to Timothy) is dumping on the concept of exercise. However, the English phrase should be read literally: he's comparing exercise to being godly. Physical exercise is good for a few things, but godliness is good for all things because of what Pseudo-Paul thinks godliness can accomplish as opposed to what physical exercise can accomplish.
1 Timothy 4:12 - In Conversation
Let no man despise thy youth; but be thou an example of the believers, in word, in conversation, in charity, in spirit, in faith, in purity.
This scripture is sometimes given to LDS youth, and in general I don't think it's a bad choice, but the word "conversation" here doesn't mean their manner of speaking. That's covered by Pseudo-Paul's encouragement to be an example "in word". Instead, conversation means their actions.
2 Timothy 3:7 - Ever Learning
Ever learning, and never able to come to the knowledge of the truth.
Mormons often apply this scripture against intellectual critics of the Church, including former members who lost their faith because they were too focused on studying the history of the organization and are now blinded from faith due to their learning. And why not, right? This chapter begins with a description of the many types of horrible people that will be around during the "last days", so obviously this verse is talking about those people.
It's not. Let's give a little more context for it from the preceding verse, which follows a long list of horrible way people will behave in the last days.
6 For of this sort are they which creep into houses, and lead captive silly women laden with sins, led away with divers lusts, 7 Ever learning, and never able to come to the knowledge of the truth.
The people who are ever learning are the people Pseudo-Paul describes as "silly women laden with sins, led away with diverse lusts". (We should note that Pseudo-Paul really is a rather blatant misogynist; there are other places in the New Testament that aren't good towards women, but Pseudo-Paul is the author of the strongest anti-women passages.) The evil people described in the scripture mastery verses lead people into ever learning, but they themselves are not described as being in this position. Pseudo-Paul never indicates if these "silly women" are Jesus followers or not. I think that to only apply this phrase of "ever learning" to intelligent people outside of the LDS Church is just a way to salve worries that their intelligence might be an indication of real problems.