Should traditional lifestyles become extinct, cultural heritages would be lost. Should traditional lifestyles disappear,cultural heritages would not be passed on for posterity. The word "extinct" is ok , and other words like "disappear" or "die out" sounds better for above. Example : Many languages are getting extinct/disappearing/dying out. Living styles is not good . It should be lifestyles, which are styles of living. You usage of "neither" is wrong : it should be "not". Should traditional lifestyles die out , cultural heritages would neither be preserved nor remembered.

This is close, but with a few corrections: 1. "Extincted" is never used. "Extinct" is used for all tenses, e.g. "The dinosaurs went extinct." 2. "Neither" is only used when referring to two items in the negative. In this case you would use "not". 3. Depending on the context, "heritages" should possibly be the singular "heritage". So your corrected sentence could read: Should traditional living styles go extinct, cultural heritage would not be carried forward.