Of course pretty much every Christian believes the Old Testament should still be part of our Bible as the shadows of Christ, still edify and prove that Jesus was the Christ:
All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness (2 Tomothy 3:16)
The New Testament is possibly not even understandable, without the gradual revelation of the Old Testament leading up to it. However, contrary to what people might think the meaning of the Old Testament 'as it should be applied today' is one of the most controversial subjects that can ever be raised among protestants. To explain the tension over the issue it is best to divide the Law into its three traditional parts and then comment on how Christians view them.
The law is commonly divided into ceremonial, moral and civil.
In Christianity it is agreed by most that Christ fulfilled the meaning of all that was ceremonial and that therefore when the curtain of the temple was torn from top to bottom these laws became 'obsolete'. From this standpoint all ceremonial laws, of which the High Priest offering sacrifices in the temple were probably the most sacred, are no longer application by obsolescence.
In the same manner most Christians believe that the civil laws that were used to govern Israel, such as the many capital punishments by stoning, burning, or strangulation ceased. These are also viewed as being symbolic of man's need for salvation under the 'curse of the Law'. Also as the invisible church of Christ replaced the literal outward church of the Jews, God no longer directly governs the Jewish people. There are some Christians who still think that God in some way governs the Jews. (I think dispensationalists still hold some ideas about this but am not sure of the details, as I am not that familiar with this unique view).
The controversy is mostly about how God’s moral law in the Old Testament should be applied today, or if there is any change to it at all. Most Christians who propose the idea that 'nothing changed' to God’s moral Law at all in Christ’s death, commonly refer to this verse where Jesus said:
"Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. (NIV Mathew 5:17)
A common verse referred to by those that believe the entire Law (moral, ceremonial, civil) was 'nailed to the cross', so that even its moral nature was modified in Christ, commonly refer to the verse:
For the law was given through Moses; grace and truth came through Jesus Christ. (NIV John 1:17).
Even among traditional reformed theologians many are divided over this question about the moral law. I am of the camp that you can’t divide the Law at all, so that all aspects were fulfilled in Christ and so nothing remains exactly as it was. For example, the Law said we should love our neighbor as ourselves, however it did not provide any ability to do so but condemned to hell any offence. In Christ however, under the liberty of grace, a stony heart has been replaced by a soft one, and an uncircumcised heart has been circumcised, therefore, Christians have love in the heart and can follow the command. Therefore, even the moral law has not gone un-crucified as it was formerly written on stone but now in a 'Spirit of liberty' it is written in a believer’s heart. In this way, Christ abolished even the moral law in the sense that he fulfilled all its requirements and then imputed that righteousness to believers, which in turn, infused the holiness required by the law as a basic 'principle of life and liberty' into Christians. In this sense the Bible say’s we are ‘no longer under law' (that is the moral law).
For sin shall not be your master, because you are not under law, but under grace. (Romans 6:14).
This last verse I quote is at the heart of the debate for depending on which theologian you consult the moral law is intended, or not intended, by this and similar verses. As for me I am absolutely convinced the moral law is chiefly intended as described above.
For details of specific theologians and their view this post explaining the difference between Biblical Theology and Covenant Theology. Biblical theology has the sole ambition of answering how the NT naturally develops from the OT. Covenant Theology carries both options about the view of the moral law. Another recent trend that has its own twist on the subject is called New Covenant Theology. I recommend reading Jown Owen’s and Luther’s view at the post, regarding how to best view the 'moral law'.