Muscatine

If you think you should be exempted from methods of birth control, since you're "morally opposed" to it...... Can I be exempted from having my taxes pay for war, which I'm "morally opposed" to?

I don't know what you mean, exactly, by "exempted from methods of birth control"?  Nor, what you mean by "war".

In the first instance, if you mean exempted from furnishing free birth control and abortion because of a religious doctrine you subscribe to, I would say "yes" you should be exempted.

In the second instance, if you can show that the religion you practice, such as Quakerism (Friends), Mennonite, etc. espouses tennants that are morally opposed to war, then I say not only should you be exempted from a portion of your federal income tax, but you also have a moral responsibility to actively work for the institution of laws to that affect.  You also have a moral responsiblity to withold a portion of your federal income tax that equals an amount that can be shown would go to support a "war", regardless of the consequences.  And I would encourage you to do so.