What should I do about my sister's refusal to celebrate birthdays and holidays?
The short answer
Point out that the Lord celebrated holidays (cf. Luke 2:41-43; John 6:3-5; 7:1-10; 10:22-23). Holidays are not worship of days or people but time set apart to focus on loving God and neighbor, the two great commandments.
We cannot presume to know your sister's interior prayer life, but you might point out that the Lord celebrated holidays during his earthly life (cf. Luke 2:41-43; John 6:3-5; 7:1-10; 10:22-23). One of the motifs of John's Gospel is the cycle of Jewish feasts.
Holidays are not for worshiping days or individuals. They set apart time to focus on loving God and neighbor, the two great commandments (cf. Matthew 22:35-40). As in secular life we set aside vacation days to rest and attend to family, in the spiritual life we must set aside holy days to attend more closely to God and neighbor. Left to ourselves, we might never do it; because the days are set apart ritually, it is already done for us.