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great question. I think we need to first define what 'love the world' means in each of these contexts (the Church lyric, John 3:16, and 1 John 2:15). In both John 3:16 and the Church lyric, to love the world is to love the lost people in the world who are in need of salvation. In contrast, to 'love the world' in 1 John 2:15 is to love the things that this world can give you more than God - the lust of the flesh, the lust of the eyes, and pride in one's lifestyle.
If we read through 1 John, as well as the rest of Scripture, we do find that we are called to love other people the same way that God loved us (see 1 John 4 below). Going back to John 3:16, we are indeed to love the world as God has loved the world. We love because He first loved us. We are called to forgive others just as God has forgiven us and to love them because He first loved us.
Now, you might argue that it is not possible for a human to love others the same way that God loves them. And in some sense that it is true - because we do not know them as God knows them. On the other hand, God's Spirit lives inside us, teaching us both how to pray and how to love others. And in that sense, being a child of God and having God live in us / give us new hearts / make us new creations does empower us to love with God's love by His Spirit.
1 John 4:7-12
Dear friends, let us love one another, for love comes from God. Everyone who loves has been born of God and knows God. 8 Whoever does not love does not know God, because God is love. 9 This is how God showed his love among us: He sent his one and only Son into the world that we might live through him. 10 This is love: not that we loved God, but that he loved us and sent his Son as an atoning sacrifice for our sins. 11 Dear friends, since God so loved us, we also ought to love one another. 12 No one has ever seen God; but if we love one another, God lives in us and his love is made complete in us.
Upvote:2
That's a really good question but it might be helpful to point out that when it says in John 3:16 "...God so loved the world", it's followed by a "that..." Read the whole sentence and we learn that his great love caused him to send his only-begotten Son into this world to be born as a human baby, who was to grow up and sacrificially die in order for sin and death to be dealt with God's way.
But your question is actually about how we can love the world the same way that Jesus loved the world. The answer to that is that the Son of God agreed to be sent into this sinful world, lowering himself from his great position in the Godhead, to become despised, to become a servant, to suffer and also to be tortured to death. All because of the immense love of the Father and the Son for fallen humanity. Therefore, for us to love as Jesus loved, we need to be willing to be sent by the Father to do his will, here below. We need to shove aside our own desires, our own ideas of our importance or our station in life. We need to be, to do, and to go entirely according to Jesus' injunction to his followers:
"Seek first the kingdom of God and his righteousness, and all these things will be added to you" (Matthew 6:33).
We can only love as Jesus loved if we have that degree of faith, and love him as our Lord, our Saviour, and the King of God's kingdom. Then we will follow the King wherever he leads. We will show all those we meet along the way that spirit of Christ, which indwells all believers (Romans 8:9). We will point others to Christ, not ourselves, and we will do that by word and by deed. That is how to show Christ-like love to the people in the world.
EDIT - Another important statement of Jesus related to what loving the world does NOT mean, is Jesus' prayer where he said:
"I pray not for the world, but for them which thou hast given me; for they are thine, and all mine are thine; and I am glorified in them." (John 17:9)
Christians likewise do not pray for the world, but for the salvation of those yet in the world who are called to come out, and for all fellow-believers.