Praise be to Allah.
It is Sunnah for the one who is doing tawaf to kiss the Black Stone in every circuit of his tawaf if he is able to do so without disturbing anyone. If it is too difficult for him to do that, then he may touch it with his hand and kiss his hand. If that is not possible, then he should point to it.
It is sunnah for him to do that in each circuit of his tawaf because of the report narrated by Abu Dawood in his Sunan (1878):
The Messenger of Allah (blessings and peace of Allah be upon him) did not omit to touch the Yemeni Corner (ar-rukn al-yamani) and the Black Stone in every circuit.
Classed as saheeh by al-Albaani (may Allah have mercy on him).
Muslim (1268) narrated that Naafi‘ said: I saw Ibn ‘Umar touching the Stone with his hand, then he kissed his hand and said: I did not stop doing this since I saw the Messenger of Allah (blessings and peace of Allah be upon him) doing it.
This indicates that it is recommended to kiss the hand after touching the Black Stone, if one is unable to kiss the Black Stone.
This hadith may be understood as referring to one who is unable to kiss the Black Stone. Otherwise, the one who is able to do so should kiss the Stone and not limit himself to touching it with his hand. End quote from Sharh Muslim (9/15).
Something similar was said by Shaykh al-Mulla Qaari concerning the phrase “touching the Stone with his hand, then he kissed his hand.” He said: Perhaps this was at a time when it was very crowded. End quote from Mirqaat al-Mafaateeh (5/1795).
Shaykh Muhammad ibn ‘Uthaymeen (may Allah have mercy on him) was asked: You said that it is prescribed for the one who is doing tawaf to kiss the Black Stone if possible, otherwise he should touch it or say takbeer. This is in the first circuit; but what is the ruling on the other circuits? What is the ruling if he does not do that?
The same ruling applies to all the circuits, but if he does not do that, it does not matter, because saying takbeer, kissing the Black Stone and touching it are all Sunnah; what really matters is tawaf (circumambulating the Ka‘bah). End quote from Majmoo‘ Fataawa ash-Shaykh Ibn ‘Uthaymeen (22/331).
And he (may Allah have mercy on him) said:
All of these actions were narrated from the Prophet (blessings and peace of Allah be upon him).
The best of them are touching the Stone with one’s hand and kissing the Stone; then touching the Stone and kissing one’s hand; then touching it with a stick and the like and kissing it, if doing so will not cause annoyance to anyone – but this Sunnah only applies to one who is riding, as far as we know; then pointing to it. So there are four possible actions, in order of preference, to be done without causing annoyance to others or putting oneself through hardship. End quote from ash-Sharh al-Mumti‘ (7/238).
If the one who is doing tawaf is able to kiss the Black Stone, or to touch it, without causing annoyance to anyone, then he should do that in every circuit, but it is not obligatory.
And Allah knows best.