Praise be to Allah.
It is only prescribed to uncover the right shoulder and walk quickly with short steps (raml) in the tawaaf of ‘Umrah, and in the tawaaf of arrival (tawaaf al-qudoom) for those who are doing ifraad and qiraan Hajj. In other tawaafs these actions are not prescribed, so in tawaaf al-ifaadah there is no raml and no uncovering the shoulder, whether you did tawaaf in ihraam or not.
Abu Dawood (2001) narrated from Ibn ‘Abbaas that the Prophet (peace and blessings of Allaah be upon him) did not walk quickly in the seven circuits of tawaaf al-ifaadah. This was classed as saheeh by al-Albaani in Saheeh Abi Dawood.
Uncovering the right shoulder and walking quickly with short steps (raml) go together. When raml is prescribed, uncovering the right shoulder is also prescribed; when raml is not prescribed, uncovering the shoulder is not prescribed either.
Al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’, 8/43
“Uncovering the right shoulder goes along with raml. When it is mustahabb [recommended] to walk quickly (raml) it is also prescribed to uncover the right shoulder, and when it is not recommended to walk quickly, it is not prescribed to uncover the shoulder either. When the scholars differed, that had to do with both uncovering the shoulder and walking quickly [i.e., the two actions are not separated]. There is no difference of opinion on this matter.
He also said: But raml and uncovering the shoulder are separate in one case, which is that uncovering the shoulder is Sunnah in all seven circuits, but raml is only Sunnah in the first three circuits, and one should walk (normally) in the last four.
Ibn Qudaamah said in al-Mughni (5/221), It is mustahabb to walk quickly and uncover the right shoulder in the Tawaaf of ‘Umrah and in the Tawaaf of arrival. Then he said:
It is not Sunnah to walk quickly or to uncover the right shoulder in tawaaf except in the cases we have mentioned, because the Prophet (peace and blessings of Allaah be upon him) and his companions only walked quickly and uncovered their shoulders at those times.
It says in Fataawa al-Lajnah al-Daa’imah (11/225):
It is Sunnah to uncover the right shoulder in all circuits of the tawaaf of arrival only, and it is prescribed to walk quickly in the first three circuits of the tawaaf of arrival for those doing Hajj and those doing ‘Umrah.
Shaykh Ibn Baaz (may Allaah have mercy on him) said:
He should walk quickly (raml) in the first three circuits of the first tawaaf, which is the tawaaf he performs when he first arrives in Makkah, whether he is doing ‘Umrah or tamattu’, or he has entered ihraam for Hajj only, or is combining Hajj and ‘Umrah (qiraan). And he should walk in the last four circuits, starting each circuit with rapid steps. And it is mustahabb for him to uncover his right shoulder throughout this tawaaf, but not in others.
Fataawa Ibn Baaz, 16/60.
And Allaah knows best.