Famously, Henry Thoreau spent the night in jail when he refused to pay a poll tax to support the Mexican War, an experience which caused him to write (and coin the term) Civil Disobedience. Later, both Joan Baez and Gloria Steinem refused to pay taxes to protest the Vietnam War (wonder what Steinem, an abortion proponent, would think of me refusing to pay my taxes because I refuse to fund abortion?)
During Vietnam, the number of Americans who refused to pay taxes, entirely or in part, grew so large that the IRS quietly decided to stop prosecuting them.
Here is a list of all the Americans who were convicted of refusal to pay taxes to support military spending since 1949, and the sentences they received. It’s less than a hundred people. Many of those were charged with filing a false or fraudulent W4. A surprising number were released after citing either their 1st or 5th Amendment rights.
nwtrcc.org/resist/consequences/war-tax-resisters-taken-court/
They could also seize your home or car, of course, and auction it to pay taxes - many of the war tax protestors purposely had few resources or divested themselves of them so they couldn’t be seized.
Here is a list of all the property that has been seized from war tax protestors since 1949. Again, it’s not large.
nwtrcc.org/resist/consequences/irs-property-seizures-war-tax-resisters/
I would also think that the political Left, if in power, would be less forgiving of a challenge to its secular authority on this issue than the Right was to prosecuting the wars.
But honestly, if even 5% of Catholic and Evangelicals refused to pay their taxes on this ground, with the support of pro-life congress members, they might have to back down.