What is the relevance of the political issues in Hamlet? Will the play be the same if we take out the plot about the war between Denmark and Norway?
In my opinion the political issues are not essential to the heart of the story. They are important in the sense of placing Hamlet in a particular time and place, but they aren't necessary for the play to work.
It's also important to remember that political intrigue within Denmark is part of the mix of motives in Hamlet, and it was common in those times for murders and plots to exist within a kingdom as people tried to gain power.
Politics may not at first glance seem all that important to Hamlet as a play, and it is true that the major points are made by Hamlet and the other major characters, not by Fortinbras or his actions. But, without the support of political ties, Hamlet would not carry the same depth as it does. The story and characters cannot exist in a political vacuum. Not only do the politics provide setting and background, they also help with characterization and make the story relevant to its outside world. For example, when the ghost of the King visits Horatio and the guards in the first scene, Horatio remarks that once the old King had slain the old Fortinbras and, in victory, taken some of his lands. He goes on to tell that the young Fortinbras is gathering up an army to (he presumes) possibly reclaim this land. Is this why the ghost is roaming? Without this apparently unstable political situation, the ghost may not have received the same attention. In addition, the reader/audience would not have been provided with this important information about the old King's heroic character which so sparks Hamlet's admiration (which has plenty to do with Hamlet's actions later on). It is interesting at the end of the play that right before he expires, Hamlet bestows the rights to the land on Fortinbras. They are both sons who have lost their fathers for political reasons: the old King killed Fortinbras' father for the land, and Claudius murdered the old King for power. Political power, that is. This ending is in many ways a convoluted reflection of the events of the past. It would be wrong to say that Hamlet is not a political play---the lust for power is a driving force in it, as is the need for political control of Denmark, as King, that sets it all in motion.