This passage occurs in the RCL Easter Season, Year B, most recently Summer of 2015.
Summary: This passage has some great beauty, but presents a great preaching challenge. First, we have some nastiness to the image: branches plucked and pruned. The Greek can soften the blow here: the words for pluck and prune also mean "lift up" and "clean." Yet, I think a real law and Gospel challenge remains: You can find all sorts of traditional discipleship tasks that connect us to God: prayer, the Word, even the community. Yet we can no more force ourselves upon Jesus than a branch can for themselves on the vine. To say to people, "You cannot abide in Jesus, so don't even try" makes a liar out of Jesus. To tell people "You just need to pray and read your Bibles" isn't totally faithful to the image here! Somehow we must invite people into abiding in Jesus while retaining the force of the image: Jesus is the root of connection, not us.
αμπελος ("vine"; 15.1) Like many metaphors in John's Gospel, a person new to the Bible can grasp its meaning, but a knowledge of the OT amplifies its significance. The OT (Hosea 14; Jeremiah 2; perhaps also Ezekiel 19, but who understands Ezekiel...) makes the claim that Israel is the vine of the Lord. Jesus here is saying "I am Israel." All the promises, all the hopes (if not the judgment) of Israel in the Bible have been transferred to Jesus.
αιρεω ("take away" or "take up"; 15:2). I thought I had a unique insight here and then I realized the NET Bible already explained in a footnote. In their words:
The Greek verb ai;rw (airoÒ) can mean "lift up" as well as "take away," and it is sometimes argued that here it is a reference to the gardener "lifting up" (i.e., propping up) a weak branch so that it bears fruit again. In Johannine usage the word occurs in the sense of "lift up" in 8:59 and 5:8-12, but in the sense of "remove" it is found in 11:39, 11:48, 16:22, and 17:15. In context (theological presuppositions aside for the moment) the meaning "remove" does seem more natural and less forced.
They actually give a HUGE footnote on this point. You can find this online through their website (bible.org) or Bible Works.
Another person familiar with vines pointed out that a non-blossoming branch must be lifted up to ensure the cut must be as close as possible. In this light, we can see that the cutting is not done far away, but hand-to-hand. When God prunes us this is done an an intimate way!
Long and short: I think for a sermon, one could introduce the idea of Jesus lifting someone up instead of simply tossing away, especially in light of this verb:
καθαιρω ("clean"; 15:2). Alas, I got this word wrong in my blog entry three years ago. I thought it was καθαριζω as in to cleanse. The two words mean essentially the same thing. However, John uses a word that allows him to have internal rhyme in a verse. More importantly, we have a very modest image, not a very harsh one, of cleansing. It is translated as "prune" only in light of the later verses.
ινα ("in order that"; 15:2) A reminder that God's cleansing and forgiveness always have a purpose! (Confessional Lutheran note: How does this cleanising happen? Through the Word of God!!)
εν uμιν ("in you"; 15:6) This can mean "in you" but it is also in the plural: "In all of you" or even "Among all of you." "Abide in me as abide among you." might be good for individualist Americans to here!
γινομαι ("occur"/"happen"/"be"/"become"; 15:7, 8) If you study this word, you will see that Jesus is not saying, "Ask for anything and it will be given unto you." He is saying, "What you ask for will be unto you." The same verb is used in 15.8, that you would become my disciples. Jesus here seems to be telling the disciples what they should be asking for -- that they would abide in him and become their disciples.
Grammar review: τις...some times the smaller the word, the more difficult to translate
The little word τις is a pain! First, depending on the direction of the accent, it can either be a question word meaning (who, what, whom, whose) or an indefinite article (a, an, any, some, one). At least this division is revealed by the accents (or lack their of; if it has no accent, it means an indefinite article because that τις has a weak accent that has been moved to the previous word)
But how to figure out then what is means is tough. In the case of verse 6, τις is universally translated as anyone. But it could just as easily be "anything." If anything is not in me, it is tossed out and burned up. This is perhaps a nicer translation. In this case, "anyone" is probably the most correct translation because Jesus has already indicated we are the branches.