Since the word "fire" is used in the preceding verses exclusively in reference to hell, then it seems unlikely that it would be used in a different way here. Literally, the text says "salted with fire." In the Scriptures, salt is a symbol of that which preserves from decay. Thus, if one is salted with fire, then he is being preserved with fire from ultimate decay (i.e., he is perpetually being kept in a state of severe pain with no end)! Jesus is stressing the fact that punishment in hell is eternal; there will be no annihilation. The word "everyone" must also be limited by the context to those who refused to "cut off" the offending part.
"And every sacrifice will be seasoned with salt" (Mark 9:49; cf. Lev. 2:13). Faithful followers of Jesus are to be "living sacrifices" (Rom. 12:1) in the way they deny themselves the pleasures of sin and live to fulfill the Lord's will. Thus, a contrast is being made in this verse between the two future states of preservation--heaven and hell. Those who are sacrifices for God will be preserved to enjoy the love of God in heaven and everyone else will be preserved in hell to endure their just punishment.
Mark 9:50 conveys a similar sentiment as Matthew 5:13. Salt preserves, but if the Christian does not persevere he is worthless (cf. Matt. 10:22; Rev. 2:10)! If we really are "the salt of the earth" then we will have "peace with one another."
Jesus then declared in Matthew 18:10-14:
"Take heed that you do not despise one of these little ones, for I say to you that in heaven their angels always see the face of My Father who is in heaven. For the Son of Man has come to save that which was lost. What do you think? If a man has a hundred sheep, and one of them goes astray, does he not leave the ninety-nine and go to the mountains to seek the one that is straying? And if he should find it, assuredly, I say to you, he rejoices more over that sheep than over the ninety-nine that did not go astray. Even so it is not the will of your Father who is in heaven that one of these little ones should perish."
Jesus resumes His warning against the pride that exalts itself and despises the humble - "Take heed that you do not despise one of these little ones" (Matt. 18:10).
"Their angels always see the face of My Father who is in heaven" - I believe the idea here is that these angels (i.e., the ministering spirits for all the saints, cf. Heb. 1:14) have constant access to the Father. For one to remain in the presence of a king (whether earthly or heavenly) indicates that the individual is deserving of high honor. Thus, since "their" angels serve them and are highly respected, then certainly all faithful disciples (even the "little ones" or babes in Christ) are honored and highly regarded before the throne of God. This verse is often used as justification for the notion of guardian angels. However, it is impossible to conclude from this text that each child of God has a specific angel of his own (without reading into the text more than is warranted). There is unquestionably a group of angels that ministers to the saints, but very little is revealed about their ways and number.
Matthew 18:11 records another good reason not to despise new converts - "For the Son of Man has come to seek and to save that which was lost." Upon this principle, Jesus makes some comments regarding sheep. A good shepherd will always search for a lost sheep and try to restore it (cf. Luke 15:4-7). Care should be given to each soul for every single one is important (cf. Matt. 16:26). Since God does not despise the erring disciple or wish that he would perish, neither should we. This should be remembered always as we labor to restore erring brethren and strive to evangelize the lost.
Matthew 18:13 is not forbidding us from rejoicing over the faithful (cf. Luke 15:31,32). The emphasis here is upon the exceedingly abundant joy that should be shared by all when one is brought back to faithfulness. The emotion of that moment of restoration should be filled with more joy than that which is continually experienced over the faithfulness of the 99. The greater joy has nothing to do with a difference in value of a particular soul (cf. Acts 10:34,35).
The section closes with the implication that it is possible for those who have become His "little ones" to perish, though God doesn't wish for this to happen (cf. II Pet. 3:9).