Are other religions treated differently by any major branch of Christianity? In Judaism, some religions are considered to be Avodah Zarah, and others are not. So, for example, if Christianity or certain of its denominations is Avodah Zarah, that has implications for Jews' interactions with those religions and with their places of worship.

Islam also distinguishes between religions which are Ahl-ul Kitab and ones which are not.

Does any major Christian tradition make similar distinctions, or does Christianity generally merely treat all non-Christian religions as non-Christian?
General orthodox Christianity believes that there is an exclusivity with Christians.  This is because their interpretation of:

John 14:6 NIV


  Jesus answered, “I am the way and the truth and the life. No one comes
  to the Father except through me.


However because of:

Mark 12:30-31 NIV


  Love the Lord your God with all your heart and with all your soul and
  with all your mind and with all your strength.’ The second is
  this: ‘Love your neighbor as yourself.’ There is no commandment
  greater than these.”


They are to have compassion for all people regardless of their belief system.  While other religions are generally considered "Pagan" or worshiping a "false" god(s).  This sometimes gets confusing when some Christians claim that another religion's main god is really Satan in disguise - which is mentioned as an enemy of God in the bible.

Judaism has a different type of the relationship with Christianity.  However its definition is not universally agreed upon.  So, is the relationship special, "yes".  Can I define it?  "No, unless you want to read a book."