I suppose a good start to get the conversation going would be the Pre tribulation rapture.
The pre tribultation rapture is the belief that before the great tribulation spoken in the end times, Jesus is going to come and snatch away those faithful Christians and spare them the torments of the tribulation. This is the dominant position held by fundamentalist today. There are many problems with this however...I guess a good place to start is a brief history.
1. It was never preached in the early Church, never had such a view been held up until 1800ish when it was invented by a man named John Nelson Darby, in a movement known as dispensationalism. Many fundamentalist will respond with a list of vague out of context references to people like St. Jerome using the word Rapture in his Latin translation of the vulgate..however keep in mind the Catholics do not deny that a rapture could happen...but it will not happen before the tribulation. Regardless, in the case of St. Jerome, he was not using the word rapture in the sense that fundamentalist use it today. So what we have, is a novel doctrine preached about 150-200 years ago, not held by any of the early Christians. It has been widely spread among fundamentalist especially with the production of the "Left behind" novels...but the funny thing is in these novels, and correct me if I am wrong, I believe the good people are left and the bad people are taken away. Showing a person who holds this doctrine that 1. it is new, and 2. it is largely popularized by a fictional book series is a good place to start in explaining this belief.
Before looking at what they base it on, let us look at the consequences of such a belief.
1. a pre tribulation rapture means that people who believe they are good Christians are expecting that they will be saved before the tribulation, they believe they will be taken up by Jesus, or raptured away..What will this do for those people that hold this view and, when the tribulation happens whenver it happens, do not get raptured? How do you think this will affect their faith? As far as they know, they were good and now they have not been raptured. What hope would they feel after this? Fear? Complete inability to collect themselves and figure out the Truth. This doctrine may seem harmless but because it is false it will have untold consequences on those who hold it at the time of the tribulation... getting it out of society now and rooting it out as much as possible will limit this type of scenario.
2. Convincing a person that there is no pre tribulation rapture will help ground them in God. If they are always focused and looking for the signs of the times, that is less time they are spending praying, reading the scriptures, and focusing on their faith. They are always looking for the next big sign, getting sucked into the next end of the world prophecy, and not doing as our Lord told us. Conversely, grounding them in Faith, helping them to understand the true meaning of suffering for Christ, and allowing them to look to the here and now and persevering in grace will give them a much better chance at salvation and open them up to the fullness of Truth in the Catholic Church.
Finally I want to point out a few key verses that this is based on..namely 1 thessolinians 4:15-16 "For the Lord Himself shall come down from heaven with commandment, and with the voice of an archangel, and with the trumpet of God: and the dead who in Christ, shall rise first. 16 Then we who are alive, who are left, shall be taken up together with them in the clouds to meet Christ, into the air, and so shall we be always with the Lord.
Taken by itself, this verse certainly sounds convincing...but the key to virtually all fundamentalist claims is that they do not take verses in context.
The context of this verse is this... at this time there were many who believed that they were going to witness the second coming of Christ.. the Thessalonians were no exception to this. The Thessalonians were concerned that those who had died were not going to be ressurected with Christ, so St. Paul was giving them words of comfort and assuring them that those who had fallen asleep were promised the same salvation and resurrection. Let us add the context of this quote, looking at verse 12 from the same chapter (4). "And we will not have you ignorant , brethern, concerning them that are asleep, that you NOT BE SORROWFUL, even as others who have no hope. 13 For if we believe that Jesus died and rose again; even so them WHO HAVE SLEPT THROUGH JESUS, will God bring with him. 14 For this we say unto you in the word of the Lord, that we who are alive who remain unto the coming of the Lord, shall NOT PREVENT THEM WHO HAVE SLEPT. and the last verse of the chapter, 17: Wherefore COMFORT YE ONE ANOTHER WITH THESE WORDs.
Once you add the rest of the verse and the context, it is easy to see what is actually being talked about here. Look at verse 16 again...we who are alive and WHO ARE LEFT...left after what? Verse 14 THE COMING OF THE LORD. After the coming of the Lord are those who are alive taken up, following the resurrection of the dead in Christ, and we will again be united in heaven.
If you take this type of approach with other verses they use...you will see similar tactics.
Always check contexts, and compare it with what the rest of the bible says. if the bible said nothing else, and it was only verse 15 and 16 it would be different. But the coming of the Lord is triumphant and glorious..not quiet, then leaving, then loud again. 2 comings..not three, or four, or even some believe in five.
Here are a couple of resources that are helpful
Catholic answers is one of my favorites.
Tim staples: former baptist pastor now Catholic apologist.