How do we identify false teachers? But false prophets also arose among the people, just as there will also be false teachers among you, who will secretly introduce destructive heresies, even denying the Master who bought them, bringing swift destruction upon themselves. -2 Peter 2:1


My question is: How can you tell if a teacher is a "false teacher"?

Please support your answer with Scripture.
1 John is really the 'discernment' Epistle showing many ways that false teachers can be identified.  The main thing is to look at the life and doctrine of a teacher to ensure they stick to the basic Christ centred ideas and life of love. The basic rules are:
False teachers do not love Christians and they live in wickedness. Real Christians know how sinful they are, confess their daily sins, but do not love the world. Real Christians love God, as proven in their love of those who believe in Christ for the forgiveness of their sins, that is God's children.

If we claim we have not sinned, we make him out to be a
liar and his word is not in us. (1 John 1:10)

The Lord's daily prayer included 'forgive us our sins' (Math 6:12). Every Christian knows that they sin every day because nobody loves God with all the heart, soul and mind, which is the greatest command. (Math 22:37)

Whoever says, “I know him,” but does not do what he commands is a liar, and the truth is not in that person. (1 John 2:4)

Some may seem correct in doctrine, but live in open sin. For example they may say it is alright to commit adultery, practice homosexuality, or the worship of someone other than God, which is idolatry, etc. Such a person is like the Devil appearing as an angel in word, but betrayed in action.

Anyone who claims to be in the light but hates a brother or sister is still in the darkness. (1 John 2:9)

A Christian cannot hate another person in the intense way that this implies. It is impossible for a born again; regenerate should to be filled with hate because God’s Spirit remains in them. This especially holds true for our attitudes towards any who belie in Christ for the forgiveness of their sins. This is the ‘brother or sister’ being mentioned. Therefore churches and cults, that consider their denomination as the only group that will be saved, are built upon false teachers. Paul says; let such teachers be ‘accursed’ even if they were a Pope, Saint, Apostle, or even an Angel. (Gal 1:8) Anyone who genuinely believes in Christ, proven by their love of all the others who also believe, will have eternal life. (John 3:16, 1 John 5:11-13)

Do not love the world or anything in the world. If anyone loves the world, love for the Father is not in them. (1 John 2:15)

Anyone who puts money, sex, career, worldly status, or religious status before the gospel loves the world and is not a Christian.  (1 John 2:16)

Whoever does not love does not know God, because God is love. (1 John 4:8)

This cuts to the spirit of all true Christians. The Holy Spirit is a loving Spirit so if that Spirit is within a soul, that soul will love others.  When somebody loves us we can tell. Not only does this help us discern our own attitudes but if somebody seems right in doctrine but is always angry, impatient, not kind, jealous of others, bragging all the time, always argumentative, selfish, or any other non-loving characteristic, the Bible seems to say we should switch channels and listen to someone else. They are clanging gongs (1 Cor Ch 13)

False teachers deny the incarnation of Christ in one way or another. They either deny the Father, Son, or Holy Ghost, or they deny the incarnation, or atoning sacrifice of sin on the cross. They might deny the whole gospel by thinking their good works will save them rather than Christ.  Real Christians believe that God assumed human nature in Christ, and that by faith in Christ alone we have eternal life.

Who is the liar? It is whoever denies that Jesus is the Christ. Such a person is the antichrist—denying the Father and the Son. (1 John 2:22)
This is how you can recognize the Spirit of God: Every spirit that acknowledges that Jesus Christ has come in the flesh is from God. (1 John 4:2)

The gospel hinges on believing that the eternal Son of God took on human nature to offer his soul and body for the punishment of your sin, and my sin. In fact faith is this one statement determines who Christians are and who liars are.

And this is the testimony: God has given us eternal life, and this life is in his Son. Whoever has the Son has life; whoever does not have the Son of God does not have life. I write these things to you who believe in the name of the Son of God so that you may know that you have eternal life. (1 John 5:11–13)

A Christian can be sure they have eternal life by relying on the death of Christ for their forgiveness. If they rely on their own works by being a good moral person, or being baptised into a specific church, they may still yet go to hell.

In the end, discernment is easy for those who have the Holy Spirit. Jesus said:

By their fruit you will recognize them. Do people pick grapes from thornbushes, or figs from thistles? (Matt 7:16)

The Bible lists the fruit of the Spirit as love, joy, peace, etc. (Gal 5:22) The nature of fruit corresponds to the type of tree from which it grows, and all Christian have been engrafted into Christ as the root, who’s life is our sap and nutrient. Naturally this fruit is not just the attitude but also the doctrine of justification in Christ by faith apart from the works of the law, as is the argument of the entire Bible.