In D&D 5e, does a successful saving throw against poison confer immunity (or at least reduced damage, depending on the poison) from that poison for life, for that combat, or only for that hit? Do I have to repeat the saving throw every time I get hit?
In general you'll need to look to the specific effect's rules. Most often you probably have something like this, from the Giant Poisonous Snake statblock:
Hit: 6 (1d4 + 4) piercing damage, and the target must make a DC 11 Constitution saving throw, taking 10 (3d6) poison damage on a failed save, or half as much damage on a successful one.
If the rules don't say otherwise, like in this case, a save protects only against that one instance of the effect - if the snake bites you again, you'll need to make the save again.
When a save protects you for longer, it'll be explicitly stated in the rules for the ability you're saving against. Often that will use wording something like this:
If the target successfully saves against the effect, or if the effect on it ends, the target is immune to this fiend's Charm for the next 24 hours.
This is actually from the Charm ability of the Incubus and Succubus, but the principle is the same - if a saving throw grants a longer-term immunity to an effect, rather than simply preventing or weakening a single instance, the ability itself will tell you so.