First of all, it should be clear that fasting the six days of Shawwal is a sunnah that is highly rewarded by Allah. Prophet Muhammad (peace and blessings be upon him) is reported to have said, “Whosoever fasted the month of Ramadan and then followed it up with (fasting) six days of Shawwal, it is as if he fasted forever” (Muslim). Being a sunnah means that omitting it is not a sinful act for a Muslim.
With regard to this issue, The Standing Committee for Islamic Research and Iftaa’ in Saudi Arabia states,
Fasting six days of Shawwal after the day of `Eid is a sunnah. It is not wajib (obligatory) on the person who does this once or more often to continue doing it. He is not guilty of a sin if he does not fast these days.