You cannot object morally to federal income taxation in the United States, although you can reduce your taxes on that income by making charitable deductions or by declining to earn income.
Certain clergy from faiths that have a moral objection to Social Security can be exempt from FICA taxation by statute rather than by any constitutional right to do so, but they give up any benefits from FICA funded programs by doing so.
In the United States could I sue the Federal government because I
don't agree with my money being used to kill people in wars?
Is there any legal precedent?
Yes. There are many legal precedents declining to allow this to be done
Are there any active movements to try to get people to do this,
Yes. They are crushed mercilessly by federal tax officials and prosecutors. Serious tax penalties, property seizures, criminal convictions and prison terms routinely follow in all cases of people who attempt to do this in practice.
or to join some kind of class action lawsuit?
Someone could attempt to bring a class action lawsuit. It would in all likelihood be dismissed before a judge certified the class as a frivolous lawsuit and would probably result in court issued sanctions against the persons bringing the lawsuit and their counsel. It would fail both on the merits and for lack of standing to sue in all likelihood.