I have seen people putting their right hand on chest or heart (I don't know exactly) after shaking hands and saying salam. Is this sunnah of Prophet Muhammad (peace be upon him) ?
No, it isn't Sunnah
Shaking hands is something that is encouraged in sharia, and it is a cause of sins being forgiven.
It was narrated that al-Bara’ ibn ‘Aazib (may Allaah be pleased with him) said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said:
“There are no two Muslims who meet and shake hands, but they will be forgiven before they part.”
Narrated and classed as hasan by al-Tirmidhi (2727); narrated by Ibn Maajah (3703); classed as saheeh by al-Albaani in Saheeh al-Targheeb, 2718.
As you mentioned, some people over their hearts/chest after greeting and shaking hands with someone.
This action is contrary to the meaning of shaking hands in both linguistic terms and Islamic terminology.
Shaking hands means clasping the palm of one hand against the palm of the other person’s hand.
Al-Raaghhib al-Asfahaani said:
Shaking hands means stretching the palms of the hands (against one another).
Ghareeb al-Qur’aan (1/282).
And Ibn Hajar al-‘Asqallaani said:
Shaking hands means clasping the palm of one hand against the palm of the other hand.
Fath al-Baari (11/54)
There is no evidence for that in the Sunnah, and there is no report – as far as we know – which says that any of the salaf did that.
So the action of putting your hands over your heart and chest is not Sunnah but probably a customary action to show honour and respect.
And Allah knows best.
Source: Islam Q&A