In Matthew 5:17-20 (which I quote below), it is not the prophecies of the Old Testament alone that are fulfilled by Christ, but the law itself is magnified by Him (Isaiah 42:21).
Think not that I am come to destroy the law or the prophets: I am not come to destroy, but to fulfill. For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled. Whosoever therefore shall break one of these least commandments, and shall teach men so, he shall be called the least in the kingdom of heaven: but whosoever shall do and teach them, the same shall be called great in the kingdom of heaven. For I say unto you, that except your righteousness shall exceed the righteousness of the scribes and Pharisees, ye shall in no case enter into the kingdom of heaven.
A few points here. In context, Jesus is clearly referring to the Old Testament law. He says clearly that every letter of the law is still valid for today (i.e. one jot or one tittle shall in no wise pass from it). If we break one of the least of the commandments in the Old Testament law, and teach men so, our reward in heaven will be lessened. And then, the righteousness of the scribes and Pharisees was that they were sticklers for the law; their outward righteousness was impeccable. Therefore we must have an inward righteousness that exceeds their outward righteousness, if we are even going to enter into heaven.
In the remainder of the Sermon on the Mount, Jesus explains how the Old Testament law is not merely outward, but how in specific ways (He is not extensive), it applies to the heart and not just outward behaviour.
Now it is also clear from scripture (Romans 8:4-7, Hebrews 8:10, Hebrews 10:16) that as believers in Christ, the righteousness of the law is fulfilled in us (Romans 8:4; Romans 5:5 w/ Romans 13:8-10) because in our minds we are subject to the law of God, being spiritually-minded (Romans 8:7); and that the law of God is written in our hearts and on our minds (Hebrews 8:10, Hebrews 10:16, Jeremiah 31:33), as believers under the New Covenant.
Now love is definitely the fulfilling of the law; but think: what this means is that if I love my neighbor as myself, I am not going to (Romans 13:8-10) commit adultery with his wife, kill him, steal from him, lie about him, covet anything that belongs to him; and neither will I break any other commandment in the law that applies to my relationship to him (Romans 13:9).
It is clear from 1 John 5:3 and 2 John 1:6 that to love God is to keep His commandments. And God is not the God only of the New Testament.
Now in the New Testament we are not required to keep His commandments for salvation, such as the Old Testament food laws (1 Timothy 4:1-6); however because we are saved we ought to consider keeping all of the commandments out of love for God. If I don't, then God is able to make me stand (Romans 14:4), as there is forgiveness in Him. But sin is defined by the law of God (Romans 3:20, 1 John 3:4), and therefore if I break any part of the law of God, I am sinning (James 2:10, Galatians 3:10). However, as a believer in Christ, I am not under the law (Romans 6:14), am dead to the law (Romans 7:4, Galatians 2:19), and am delivered from the law (Romans 7:6); and therefore the law no longer points me out as a sinner when I violate the law of God (Romans 4:8, Romans 5:13, Romans 4:15 w/ 1 John 3:4; in conjunction w/ Romans 6:14, Romans 7:4, Romans 7:6, Galatians 2:19, Ephesians 2:15-16, Colossians 2:14, and Hebrews 7:18-19). Nevertheless I am governed from the inside by the principles of the law (which amount to the love of God) (again, Hebrews 8:10, Hebrews 10:16, Romans 8:4-7, Romans 5:5 w/ Romans 13:8-10); while I am not condemned from the outside by commandments written on stone (Romans 4:8, Romans 6:14, Romans 7:4, Romans 7:6, Ephesians 2:15-16, Colossians 2:14, Hebrews 7:18-19). Rather, the commandments are written on the fleshy tablet of my human heart (2 Corinthians 3:2-3). The love of God is shed abroad in my heart through the Holy Ghost (Romans 5:5) because of my faith in Christ (Galatians 3:14); and this love, worked out in my life not in word or in tongue only, but in deed and in truth (1 John 3:17-18), is the fulfilling of the law: being a supernatural work that the Lord does in my heart. He also causes me to walk in His statutes and in His judgments because I am born again (Ezekiel 36:25-27). He works within me both to will and to do according to His good pleasure (Philippians 2:13).