Is the Israel of the NT the modern country? The Israel talked about throughout the Old Testament is fairly clearly a specific people group and later a country. The New Testament also makes references to Israel. Are these references comparable? Are they still talking about the Jewish people group and modern nation of Israel?

Is there any reason to believe that the NT mentions of Israel are to be understood differently?

As an example, consider this verse fragment:


  Romans 11:26a (ESV)
  And in this way all Israel will be saved...


Should this verse be understood to mean that all the citizens of the modern country of Israel will be saved or is the sense different? If different, where does this difference start and how is one to know when a reference to 'Israel' carries which meaning?
Not at all. Israel itself has many meanings in the OT:


Israel = Jacob, son of Isaac  
Israel = The nation of Israel (all 12 tribes of Israel)  
Israel = The northern tribes of Israel (as opposed to Judah)  
Israel = The people of God


The most common usage in the Old Testament is that it's referring to the nation of Israel (the full nation until Northern Israel went into Exile and then Judah). But although Israel were the people of God, many of them weren't really. We see this because of their sinful rebellion and God punishes them through two exiles. Their hearts were hard and they frequently chased after foreign gods.  

And yet, we do find a portion of Israel, called the remnant or True Israel, who serve God as the nation were supposed to. We see this when Elijah is hiding after having slayed the priests of Baal and God tells him that he has preserved 7000 people faithful to him (1 Kings 19:18). In fact, all through the Old Testament, the idea of the faithful remnant is present. The remnant is really what the whole nation was supposed to be (a people who loved God and devoted themselves to honouring Him).

And so we come to the new Testament. Paul gives a great teaching on this in Romans: 


  Romans 9:6-8 (NIV)  
  &nbsp;6 It is not as though God’s word had failed. For not all who are descended from Israel are Israel. 7 Nor because they are his descendants are they all Abraham’s children. On the contrary, “It is through Isaac that your offspring will be reckoned.” 8 In other words, it is not the children by physical descent who are God’s children, but it is the children of the promise who are regarded as Abraham’s offspring.


He makes the distinction I've explained above - not all Israel are Israel, i.e. not all of the nation of Israel are God's people (see one meaning of Israel applied to the first reference and another to the second). God has always kept Himself a faithful remnant and indeed, this is how it is for us as Christians. God's People (the New Israel) who are defined as people who accept God by faith, regardless of whether they are Jew or Gentile (Romans 3:21-31).

So returning to your verse:


  Romans 11:25-26a (NIV)  
  &nbsp;25 I do not want you to be ignorant of this mystery, brothers and sisters, so that you may not be conceited: Israel has experienced a hardening in part until the full number of the Gentiles has come in, 26 and in this way all Israel will be saved.


Christians everywhere are the new Israel, the People of God, and they comprise Jews and Gentiles. It has nothing to do with the modern country of Israel.