I was honorably discharged from the military in 2003.  Two years
later, I was recalled to service to fight in Iraq.  I have refused to
obey the orders due to philosophical opposition to the war and my
dedication to my studies as an undergraduate.  This will
likely result in me receiving an other than honorable discharge.  Will
this second discharge affect my chances of being admitted into a good
law school, even though I otherwise have a spotless and exemplary
record--high 160s LSAT and 3.97 GPA?

I don't know what effect an other-than-honorable discharge might have
on your law school admission chances, but it does sound like the kind
of thing you'd have to report and explain to the state bar when you
apply for admission afer law school.  I'm sure it varies from state to
state, but in California you'd definitely have to report something
like that as part of your Application for Determination of Moral
Character.

If you are planning on law school you should realize the it is all about process.
Have you formally notified the military of your objections?  You
signed a contract and if you still have obligations under that
contract or the law I don't know if a law school would admit you, let
alone the bar.