Praise be to Allah.
You do not have to bury them. If you do bury them that is good, but if you do not do that, there is no blame on you, if Allah wills.
There are hadiths which speak of the command to bury hair and nails, but they are not saheeh (sound).
Al-Bayhaqi said in Shu‘ab al-Eemaan: A hadith has been narrated about burying hair and nails via several isnaads, but they are all da‘eef (weak).
Nasb ar-Raayah fi Takhreej Ahaadeeth al-Hidaayah (1/189).
Imam Ahmad ibn Hanbal (may Allah have mercy on him) said: Hair and nails may be buried, but if someone does not do that, we do not think that there is anything wrong with that.
Narrated by al-Khallaal in at-Tarajjul (p. 19).
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) was asked about the ruling on burying hair and nails after cutting them.
Some of the scholars stated that burying hair and nails is better and more appropriate, and that was narrated from some of the Sahaabah (may Allah be pleased with them). As for the idea that throwing them away without burying them is a sin, that is not correct. End quote.
Majmoo‘ Fataawa ash-Shaykh al-‘Uthaymeen (11/answer no. 60).