If one of my opponent's creature with lifelink deals damage and that opponent forgets to adjust his life total, do I need to tell him? Consider that he realizes his mistake the turn after. If we call a Judge will the Judge let him adjust his life total?

This happens in a modern tournament, Regular REL.

Since lifelink is no trigger, yes both players are responsible for remembering to adjust life totals. So yes you need to tell him.

Thanks for the precision and please forgive me for that little ruling error of mine !