No.

Unexplored areas will spawn in any new content to that area when you first explore them. In rare cases where a world is heavily explored (not map revealed, actually explored on foot) it might be best to restart.

Though in that case, if you are up for trying one mod one time "Upgrade world" mod is one I have used to keep my original (IE started 800 hours ago) world rejuvenated with all new content to this very day

No, you don't need to start new world seed to see the new updates if you didn't explore areas with unfinished biomes. Landscape of an area generates as soon as you come closer than 500m to it, but last time they changed elevations and biome generation. You can check difference between old and new versions by entering your world's seed: https://valheim-map.world/