I just started reading about Feynman diagrams in Griffith's Introduction to Elementary Particles. Using the convention that time goes to the left, and arrows pointing forward in time represent particles and those pointing backward represent antiparticles, what does it mean if a particle points straight up, i.e. at a constant time?
It means nothing.
If you want to look at such a diagram as a diagram of worldlines (as you seem to be implying), this isn't what a Feynman diagram is. A Feynman diagram is a graphical object that encodes certain integrals in perturbative quantum field theory. The lines in it are not worldlines of actual particles, and to say that "time goes to the left" is totally meaningless. This is exacerbated by the fact that for computational purposes one usually prefers to draw those diagrams in momentum space where there is no "direction of time". The only meaningful thing that "time goes to the left" tells you is that the lines at the left end represent the input particles and the lines at the right end represent the resulting particles of this scattering process.