Do Christians still have to obey the Old Testament law?

Yes Christians still have to uphold the Law. Jesus did not come to abolish the law, but to fulfill it. 

The Jews were trying to fulfill the law through their own interpretation which is by works. But we can only fulfill the law through accepting Jesus’ interpretation of the law which is done so by faith. 

Romans 10:3-4 – Since they did not know the righteousness of God and sought to establish their own, they did not submit to God’s righteousness. 4 Christ is the culmination of the law so that there may be righteousness for everyone who believes.

No one can fulfill the law of God themselves. They must adopt Jesus’ teachings and interpretation in order to uphold the law.

James 2:17 – In the same way, faith by itself, if it is not accompanied by action, is dead.