What guidance has Scripture or church fathers offered on how to deal with sacred objects of other religions? I've heard of a case of a Christian tearing up a prayer booklet of another religion in a somewhat violent manner.  Is this action in tune with Christian tradition or conflict with any biblical edicts?
My answer has to do with an assumption that the prayer book of another religion can essentially be viewed as an idol, given that some may attempt to get something "spiritual" from an object constructed by humans.  And here are some verses that apply to answer your excellent question:

1) Don't bow to the idol.
There are degrees of reverence to the object  that those in other religions may deem necessary (one can imagine other religions bowing to it-- for example, I've seen a Catholic priest bow to the Bible during a worship service). Biblical principle: don't participate.


  "Yet I will leave 7,000 in Israel, all the knees that have not bowed to Baal and every mouth that has not kissed him." 1 Kings 19:18


2) Tactful destruction of a tract may be biblically warranted.
There are biblical cases where idols of other religions were destroyed (2 Kings 19:18).  For example, I have picked up a free music CD from a store, played it at home, found it to contain violent sexual lyrics, and then quietly broke CD in half and thrown in garbage.  I have also quietly pocketed and trashed a tract left out in public for something I found spiritually harmful.

3) The idol is mere wood.
Treating a book as a mere book might be a better witness than an elaborate display of going after it like you need to stop its spiritual power from infecting everyone. 


  And have cast their gods into the fire, for they were not gods, but the work of men's hands, wood and stone. Therefore they were destroyed. 2 Kings 19:18 


4) Physically attacking symbols of religion may be less effective than witnessing to the foolishness of the religion. 
Paul waded into the midst of idols an gave sermons where he told those in false religions about the true religion.  The text does not record him knocking down the idols.
Similarly, I cannot count the number of times my pastor(s) have given detailed instruction on other religions. To be clear, kids who didn't know about other religions, knew more about these religions after the class.  So information about other religions isn't necessarily harmful. 


  For as I walked around and looked carefully at your objects of worship, I even found an altar with this inscription: to an unknown god. So you are ignorant of the very thing you worship—and this is what I am going to proclaim to you. Acts 17:23


5) Mockery, incisive Socratic questions are okay.
Jesus mocks the Pharisees' religion with several unflattering analogies in Matthew 23.