If someone blocks an infect creature, do the -1/-1 counters stay on the creature?
They stay on the creature. The rules for Infect say:


  702.89c Damage dealt to a creature by a source with infect isn’t marked on that creature. Rather, it causes that many -1/-1 counters to be put on that creature. See rule 119.3.


There is nothing in the rules that removes them, therefore they stay on the damaged creature.

From the Gatherer rulings on Vector Asp:


  The -1/-1 counters remain on the creature indefinitely. They're not removed if the creature regenerates or the turn ends.