What does John 13:8 mean?

Peter objects to Jesus performing the submissive task of a servant: washing of others' feet (John 13:6). His motivation is noble, since he clearly sees Jesus as his master. In the prior verse, Jesus reassured the disciples that they would better understand this action later (John 13:7).

Peter, never quick to take a hint, responds with a strongly-phrased expression in Greek. His exact words are "ou mē…eis ton aiōna!" This is somewhat like saying "no way…not ever!" in English. This is sincere, but stubborn. Peter does not understand, but suggests he doesn't need to: he cannot imagine Jesus acting like a servant. If he can't understand it, he won't cooperate with it.

Jesus' response is equally black-and-white. His wording is an ultimatum: either I do this, or you have nothing to do with me. In the most immediate sense, this is a rebuke of Peter's stubbornness. So far as the literal, physical foot-washing between Jesus and Peter is concerned, it's a non-negotiable. One can imagine Jesus saying something to the effect of "I'm not asking you Peter, I'm telling you."

Scholars differ in how—or if—to apply Jesus' comment to believers in general. On one hand, there's a universal sense in which those who abjectly refuse to submit to Christ are, by definition, not His followers. If someone follows through on an attitude of "I am not doing that, Jesus" then they have "no part" in Him or His kingdom. On the other hand, this comment can't be separated from the literal context or the statements Jesus is about to make.

As the passage continues, Jesus will distinguish between "washing" and "bathing." This will include an allusion to Judas' betrayal. That background makes an application of Jesus' words to the modern believer more specific: this speaks about fellowship, not eternal salvation.