Matt 1:22-23 cites Isa 7:14 as being fulfilled by Jesus being born of the virgin Mary. See The mythological birth of Jesus for an analysis of this passage.

Matt 2:5-7 says Micah 5:2 is fulfilled by Jesus being born in Bethlehem. While Matthew has Bethlehem as Joseph and Mary's home, Luke says that they only travelled to Bethlehem because of a census (Nazareth is actually called 'their own city' in Luke 2:21-22,39). In fact the Micah text relates to a renewal of Judah and clearly relates to David (see 1 Samuel 16:1, 17:12) and is not relevant to a future messiah. Micah is also misquoted.

Matthew 2:15 cites Hosea 11:1 as being fulfilled by Jesus and his parents leaving Egypt and living in Nazareth. But the Hosea text is not a prophecy as it relates to the story of God calling the Hebrews out of Egypt in the exodus: the Hebrews are call God's 'first-born son' in Exodus 4:22.
Furthermore a stay in Egypt is not only omitted in Luke but also made impossible (See The mythological birth of Jesus).

Matthew 2:17-18 says Jeremiah 31:15 is fulfilled by the slaughter of male infants, ordered by Herod. The Jeremiah text actually refers to the writer imagining Rachel, the mother of Joseph and Benjamin, weeping from her tomb in Ramah over her slaughtered children; the text then has God telling the exiles not to weep as they would one day return.
There is nothing in the text which refers to a wholesale slaughter of infants in the future. Note that Josephus, the Jewish historian, who was keen to record undesirable details about Herod, makes no mention of it. It is not mentioned in any other New Testament writing.

Matthew 2:23 refers to an Old Testament prophecy being fulfilled by Jesus living in Nazareth. As observed by commentators, the Old Testament text 'quoted' by Matthew's author does not actually exist.

Matthew 3:3 refers to Isaiah 40:3 being fulfilled by John the Baptist 's ministry. Firstly, Matthew's author correct Mark's error of quoting Malachi 3:1 and Isaiah 40:3 and referring to them as just 'Isaiah'.
Secondly, Matthew deliberately misquotes the Isaiah text to agree with what is said about John the Baptist (i.e., preaching in the wilderness). In fact Isaiah 40:3 does not say: 'The voice of the one crying in the wilderness: "Prepare the way of the Lord"' (as Matthew quotes it), but 'A voice cries: "in the wilderness, prepare the way of the Lord"', i.e., the voice was not in the wilderness, but speaking about the wilderness. Furthermore, there is no suggestion in the Isaiah text that it is referring to the forerunner of a coming messiah.

Matthew 4:12-17 cites Isaiah 9:1-2 as being fulfilled when it describes Jesus' movements in the early part of his ministry. In fact Isaiah 9:1-2 relates to the restoration of the Davidic kingdom when the oppressors will be overthrown and the ideal king will rule over the restored kingdom; the people await a divine warrior who will destroy their enemies and establish an everlasting rule (A description of this king is given in 2 Sam 23:3-7 and Isaiah 11:1-16).
The description supplied in Isaiah 9:3-7 which follows the Isaiah 9:1-2 prophecy does not agree with the Gospel record of Jesus' life and it appears that it has only been quoted because it mentions some locations which Matthew mentions in 4:12- 13.

Matthew 8:16-17 cites Isaiah 53:4 in respect of Jesus healing the sick. However this text refers not to the messiah, but the people of Israel. In fact, the passage is usually cited by Christians as a prophecy of Christ's suffering and death, but here, Matthew 8 uses the text to refer to Jesus healing the sick.

Matthew 12:15-21 refers to Isaiah 42:1-4 when it describes Jesus healing the sick and ordering those who are healed and are following him to be silent about what they had seen. However, the text is not relevant to Jesus as it refers to the people of Israel.
Secondly, Matt actually quotes the words that say 'and he shall proclaim justification to the Gentiles' (12:18), but this is not appropriate to Jesus whose mission was to Israel and him telling his disciples to 'Go nowhere among the Gentiles' (Matt 10:5).
Furthermore, Matt also quotes the words 'He will not wrangle or cry aloud nor will any one hear his voice in the street' (12;19) but according to the Gospels, Jesus did cry aloud and publicly preach in the towns and cities (e.g., John 7:37). The Isaiah text also says that 'he' will not fail or be discouraged but according to the gospels, Jesus was discouraged to the point of utter despair (e.g., Mark 15:34). Thus, the Isaiah text was not 'fulfilled' by Jesus.
In fact, as one of the 'Suffering Servant' poems in Deutero-Isaiah, Isaiah 42:5-9 refers not to a messiah, but the nation of Israel and asserts the justice that will prevail in Israel will make it an example to other nations.

Matthew 13:13-15 refers to Isaiah 6:9-10 being fulfilled when it describes Jesus using parables that could not be understood. In fact the text in Isaiah relates not to parables or a coming messiah but God speaking to Isaiah and telling him the warning of coming disaster will be 'heard' by 'deaf ears'. The text is unclear and the Dead Sea MSS has different wording in some places, but it appears to relate to Israel in 722 BCE and then (in 6:11-13) to the people of Judah.

Matthew 13:34-35 uses Psalm 78:2 in respect of Jesus speaking in parables. It should be noted that some MSS have 'spoken by the prophet' and others have 'spoken by the prophet Isaiah', but both are wrong as the quotation is from Psalm 78.
In this psalm the writer refers to how the traditions, which are God's testimony, were passed down from father to son. The text then goes on to speak about God's power. There is nothing in the text which relates to the future messiah teaching in a way that he could not be understood, i.e., parables.

Matthew 15:7-9 relates to the hypocrisy of the scribes and pharisees and cites Isaiah 29:13. But the Isaiah text deals with the failure of the people, being part of the material relating to disaster (29:16-16) and deliverance (29:17-24) for Jerusalem. The statement is in the present tense, and not the future, and therefore cannot be prophetic or messianic.

Matthew 21:1-5 sees Zechariah 9:9 as a 'prophecy' fulfilled when it describes Jesus entering Jerusalem. Here Matthew's author displays his ignorance of the Old Testament and Hebrew. He understands Zechariah 9:9 to refer to two animals and therefore writes of two animals being involved in Jesus' entry when the Zechariah text actually only mentions one. The text is an example of Hebrew parallelism in which lines were matched by sense rather than sound. The Old Testament abounds with synonymous, antithetic, synthetic and cumulative parallelism - repetition, contrast, and extension. The other three Gospel writers did not misunderstand the parallelism of the Zechariah text and only have one animal involved in Jesus' entry into Jerusalem (Mark 11:2-7, Luke 19:3-35, John 12:14-15).
The Zechariah passage does refer to the coming king and all the four Gospel writers no doubt felt obliged to use it, but Matthew's error over the Hebrew parallelism betrays the fact that the writers deliberately composed or shaped stories to agree with what they considered to be Old Testament prophecies.
There is the further point that the passage goes on to say (9:10-12) that the king will bring about global peace; this hardly applies to the New Testament Jesus or those who claim to follow him.

Matthew 21:33-43 uses Psalm 118:22-23 and apparently relates it to the Gentiles as the story told is about the Jews losing their special place and this being given to another nation or people (21:43,45). However, when Psalm 118 is cited in Acts 4:11 and 1 Pet 2:7, it is clearly meant to refer to Jesus rather than the Jewish people.
In fact Psalm 118 is a song of thanksgiving for the Israelite king's victorious return from battle; when he arrives at the temple gates, he asks to be allowed entry (118:19), and the response is that only the righteous may enter (118:20). The king then replies that his victory is a sign of God's favour. This is followed by the chanting of praise in the temple (118:26,27).
118:22-23 refers to the temple building stone, which although scorned by the other nations, will made important. The Psalm is a hymn of celebration and thanksgiving. There is nothing in the text which can relates to a future messiah (the interpretation of Acts and 1 Peter), and in view of the context, i.e., a hymn of celebration, it certainly could not refer to God's favour being transferred to another people Matt's interpretation).

Matthew 26:31 cites Zechariah 13:7 in respect of Jesus being arrested and his disciples deserting him. The passage in Zechariah is admitted as being complex and 13:7 may relate back to a previous reference about a shepherd in 11:3- 17.
In 11:8 it is stated that God has destroyed three shepherds and that the people reject God's shepherd and God therefore abandons them. In 11:17, the shepherd is smitten for deserting his flock, but in 13:7, the shepherd is smitten to effect the scattering of the flock. The passage following says that two-thirds will die and the remaining third will be purified through fire. Clearly none of this can relate to Jesus or the episode in which Matthew's author uses it. There is nothing in the passage that can be seen as referring to a future messiah or his followers deserting him.

Matthew 27:3-10 uses Zechariah 11:13 to refer to Judas returning the thirty pieces of silver that he was paid for betraying Jesus. As stated above when discussing Matthew 26:31/Zechariah 13:7, the quotation occurs in the part of Zechariah where the meaning is acknowledged as being obscure.
The passage appears to describe God's shepherd being rejected and he therefore abandons those responsible to the 'sheep traffickers' who recognize what is happening. The shepherd, who is really God, receives wages of thirty pieces of silver which is thrown into the temple treasury (the Hebrew word for 'treasure' is similar to 'potter'). God then declares in 11:15-17 that a shepherd will arise who will not care.
In the passage, it is God who receives the silver; even if this widely accepted view is not accepted, it is not a traitor or betrayer who receives the silver but one who protects and makes a covenant with the people (11:10-11,14). This obviously cannot apply to Judas Iscariot.
It should be noted that Matthew's author makes yet another mistake when he says the text is from Jeremiah: in fact it comes from Zechariah.

Matthew 27:35 uses Psalm 22:18 in respect of the gambling for Jesus' clothes when he is crucified. Psalm 22 is a lament of total despair, beginning with a cry that God has forsaken the writer who then reflects on God's dealings with Israel (22:3- 5), and himself (22:9-10). He continues by saying that his enemies rejoice over his despair and how he is suffering (22:6-8,11-18), and calls upon God for help (22:19-21). The writer promises to give public testimony and says that all will worship God (22:22-31).
The text used in Matthew is when the author of Psalm 22 describes his feeling of being attacked and his attackers count him as one already dead by taking his clothing and dividing this up by casting lots (22:12- 21).
In sum, as Psalm 22 describes a situation of despair the Gospel writers considered it appropriate when describing Jesus' passion but there is nothing messianic or prophetic in it and the events described have no relationship with those described by the Gospel writers.