If you're white or male, no one will expect you to go as far as John Howard Griffin or Christine Jorgensen. Most people will simply expect you to treat everyone as your social equal. Even among privilege theorists, you needn't do more than periodically acknowledge that you're aware of the privileges you have being white or male—you're not expected to give up those privileges because it's assumed you can't give them up without expensive medical aid.
But if your privilege is economic, everyone knows it's possible to give it away. Jesus, Buddha, and Moses did, and Mohammad chose to live very simply.
Talk about race or sex, and some people will feel guilty for things they can't change. But talk about class, and some people will feel guilty for things they could change.
P.S. Because I hate making people feel guilty, I'll note that while I admire saints, I agree with Marx and Engels: what matters is creating a society in which everyone has the resources for what the Declaration of Independence promised, life, liberty, and the pursuit of happiness.