Do you know of any articles that address the true dangers of teachers from outside the church vs. those that are respected teachers from within the church, that teach false doctrines? It seems that there are some who are teaching that one can not even shake the hand of a person who arrives at your door if you suspect that they might be a teacher of a false doctrine such as a Mormon or a Jehovah's Witness. I know of some that have gone so far as to say that shaking of their hand would be sinful. The reference made is from II John 10-11.
It seems that most of the modern translations of the Greek word 'CHAIRO' translate it into 'Greeting', whereas older translations, King James Version and earlier, stay with the departing clause of salutation as "nor bid them God speed". When I look up the word Greek word CHAIRO in Strongs, he states that it is a salutation of 'meeting or parting'.
I find that A. T. Robertson's (1930 publication, Word Pictures In The New Testament) seems to say that here in II John 10 "here the point turns on the stranger bringing into the house (or trying to do so) his heretical harmful teaching which seems to be after the salutation is over." - - - -
In this setting it would seem that the greeting is over with, a discussion followed, and departing remarks are to be made with the warning of 'don't send them away with the understanding that you or God supports or accepts their teaching as truth.'
I realize that this seems to be bordering on nitpicking, but how can one ever teach a false teacher what is right if we take the approach that 'I won't shake your hand' as I greet you because you may try to teach me a false doctrine, and I don't want to sin by my shaking your hand?
"Whoever transgresses and does not abide in the doctrine of Christ does not have God. He who abides in the doctrine of Christ has both the Father and the Son. If anyone comes to you and does not bring this doctrine, do not receive him into your house nor greet him; for he who greets him shares in his evil deeds" (II John 9-11).
The word translated as "greet" in this passage is the Greek word chairo, which is defined as "rejoice, be glad, welcome, greetings." The word served in the Greek as a greeting, such as "good day," "welcome," or "good morning" does in English. You can see this in James 1:1, "James, a bondservant of God and of the Lord Jesus Christ, To the twelve tribes which are scattered abroad: Greetings." Or Matthew 28:9, "And as they went to tell His disciples, behold, Jesus met them, saying, "Rejoice!" So they came and held Him by the feet and worshiped Him." It is more than just a greeting, as Paul uses it, "What then? Only that in every way, whether in pretense or in truth, Christ is preached; and in this I rejoice, yes, and will rejoice" (Philippians 1:18).
The basic point this that when a person teaches a false doctrine, a faithful follower of Christ does not leave the impression that they are in fellowship with the false teacher. I vividly remember once talking with a member of a Pentecostal church about whether miracles continue today. I showed him solid evidence from the Scriptures that God said they would cease, but he would not accept it. He preferred to believe that miracles continued, even though he never saw one and only heard rumors of rumors that they might exist. Despite our strong disagreement, he told me as we walked out the door, "the Lord bless you in your work." I couldn't help thinking, "You can't really mean that!" We had nothing in common in our view of the Bible.
What you are really asking is, is it wrong to be polite to those who come to the door, even though you know they do not represent the truth. There is a second word in II John 9-11 should be considered. We are warned not to "receive him into your house." The Greek word lambano, translated as "receive" means "to take hold of, grasp, seize, receive, get, or obtain." For example, in Matthew 5:40, the word is used for allowing someone to take your shirt and coat in court. In Luke 20:29, it refers to taking a wife. But in the case of II John 11, we are dealing with matters of doctrine and fellowship. "Most assuredly, I say to you, We speak what We know and testify what We have seen, and you do not receive Our witness" (John 3:11). "And what He has seen and heard, that He testifies; and no one receives His testimony. He who has received His testimony has certified that God is true" (John 3:32-33). It is in this latter sense that I think John is pointing out. We should not give an indication that we accept a false teacher's doctrine, even indirectly.
In our culture, we shake hands, even with our enemies. It is considered polite, but it doesn't indicate whether we agree with them. Having a person in as an overnight guest would be an entirely different matter. Giving a person a neutral "hello" doesn't tell your opinion, but waving as they walk out and saying, "have a blessed day" would indicate more agreement than would be appropriate. We should be polite, but we should not, in our politeness, indicate approval or acceptance of a doctrine that we know to be false.
Balancing these points is that fact that the false teachers under consideration are those passing themselves off as faithful brethren. In some sense, you could call an idolator a false teacher. "I wrote to you in my epistle not to keep company with sexually immoral people. Yet I certainly did not mean with the sexually immoral people of this world, or with the covetous, or extortioners, or idolaters, since then you would need to go out of the world. But now I have written to you not to keep company with anyone named a brother, who is sexually immoral, or covetous, or an idolater, or a reviler, or a drunkard, or an extortioner--not even to eat with such a person. For what have I to do with judging those also who are outside? Do you not judge those who are inside? But those who are outside God judges. Therefore "put away from yourselves the evil person"" (I Corinthians 5:9-13 ). The fact that we live in the world means we are going to have contact with people who do not believe in Christ as we do. You are going to have co-workers who are of another religion. You will likely sit down at lunch with them and you will give them greeting because it is the polite thing to do. The prime difference is that the ties of association are based on business, family relationships, or location in your neighborhood. Just because you live next door to someone, people don't conclude ideas about your religious stance. Just because you have business dealings with a person, people don't draw conclusions about the similarity of your beliefs.
As you noted, we are required to go out into the world to teach people the gospel. This will require contact with those with false beliefs -- if they had an accurate belief, there would be no need to teach them. Recall the Pharisees charge against Jesus: "Now it happened, as Jesus sat at the table in the house, that behold, many tax collectors and sinners came and sat down with Him and His disciples. And when the Pharisees saw it, they said to His disciples, "Why does your Teacher eat with tax collectors and sinners?" When Jesus heard that, He said to them, "Those who are well have no need of a physician, but those who are sick. But go and learn what this means: 'I desire mercy and not sacrifice.' For I did not come to call the righteous, but sinners, to repentance."" (Matthew 9:10-13). Jesus' association with sinners, sitting down to eat with them, did not indicate that he agreed with their sinful lifestyle. In the same manner, Jesus had many strong disagreements with the Pharisees of his day, but he did eat in the homes of Pharisees (Luke 7:36; 11:37; 14:1).