According to Jafari Fiqh reading Quran is not Fard (Obligatory) but it is highly recommended (Mustahab Mu'akkad).
In the Shari'ah (Islamic Constitution) deeds and actions are divided
into five classes:
Fardh or Wajib: فرض او واجـب An obligatory duty the omission of
which is Islamicly punishable.
Mus'tahab: مسـتـحب An action which is rewarded, but whose omission
is not punishable.
Mu'baah:: مـباحAn action which is permitted but legally is
Mak'rooh: مكـروه An action which is disapproved by the Shari'ah but
is not under any penalty.
Haram: An action which is forbidden, and Islamicly punishable.
Reading Quran is an Action and Sunnah means practices of prophet Muhammad (PBUH) that can be Wajib or Mustahab or Mubah. (Prophet was infallible and never did any Makruh or haram). sometimes Sunnah is used as meaning of Mustahab but this is general and not technical.
The Messenger of Allah (blessings of Allah be upon him and his family)
has said: “The best of those amongst you is the one who learns the
Qur’an and then teaches it to others.”
Al-Amali of Shaykh at-Tusi, Volume 1, Page 5
Imam Ja’far ibne Muhammad as-Sadiq (peace be upon him) has said: “The
Qur’an is the trust of Allah (given) to His creations, therefore it is
desirable for every Muslim to look at this trust and to recite (a
minimum of) 50 ayat (verses) of the Qur’an everyday.”
Usulul Kafi, Volume 2, Page 609
Imam Ja’far ibne Muhammad as-Sadiq (peace be upon him) has said that:
“Reciting the Qur’an from the pages of the Qur’an (meaning to look at
it and recite it - not from memory) lightens the punishment of one’s
mother and father, even if they are both disbelievers.”
Usulul Kafi, Volume 2, Page 613
References and more info about reading Quran:
40 Hadith - The Noble Qur'an