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 in Ramadan and then followed up with fasting six days of Shawwal, it is as if he fasted forever" (Muslim). Being a sunnah means that it is not sin for a Muslim who used to observe fasting these days each year not to fast them.
With regard to your question, 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.
And Allah is the source of strength. May Allah bless our Prophet Muhammad and grant him peace.
Moreover, the prominent Muslim scholar and da`iyah Zeinab Mostafa adds,
I commend you for your interest and keenness on observing the Sunnah of the Prophet (peace and blessings be upon him) in fasting the six days of Shawwal. You should know that fasting on those days is a sunnah, which means that it is not mandatory but it is highly recommended and there is a great reward promised by the Prophet (peace and blessings be upon him) for observing the fast. Therefore, it is optional to repeat this practice or discontinue it, but it is preferred to continue fasting every year and to maintain a high level of iman (faith) after Ramadan.
In his response also, Dr. Muzammil Siddiqi, President of the Fiqh Council of North America, states the following:
Fasting six days of the month of Shawwal is mentioned in a Prophetic hadith. The Prophet (peace and blessings be upon him) is reported to have said: "Whosoever fasted in Ramadan and then followed up with six fasting days of Shawwal, it is like fasting every day." (Muslim)
The meaning is that the reward is like the reward of a person who is always in fast every day of his/her life.
It is highly recommended to fast six days of the month of Shawwal, but it is not obligatory. Those who want to fast can fast after `Eid Al-Fitr any six days during Shawwal.
It is not required to fast six days continuously without any interruption. One can fast according to convenience any time during the month.