As the title says

No, time doesn't pass in the game.

Alright, then what determines the aging?

randomness at character creation
dying

The instruction manual for Wiz 7 says that time passing will age your character, although I haven't noticed anything yet.

But for Wiz 6 - rest to your hearts content.

Resting takes 8hrs of time, so your party will "age" 8hrs per rest. 3 rests equals 1 day so you'd have to rest an awful lot to add years to your party ages.

some damage over time effects are death to rest with but nobody dies of old age

Even if a character with 3 Vitality loses Vitality at level up? (This is how it worked in Wizardry 1-5 (excluding 4, as usual), and had the consequence that, if you get extremely unlucky (or use something like tool assistance to do this on purpose), a young character could die of old age.)

Maybe somebody should check this by hacking a save, setting a character's VIT to 3 and age to something high.