Question:
How is it to greet a person with salam and to stretch your hands to meet him if he is engaged in the recitation of the Quran and tasbeeh etc. in the Masjid?
N.B. Our Imam always shakes the hand of a person before the salah in answer and sometimes after it .
Answer:  حامدا ومصليا ومسلما
One should not say salam or shake the hands of a person who is occupied in zikr, tilawah and tasbeeh in the Masjid. It is makrooh to do such.
It is mentioned in Durrul Mukhtaar:

سلامك مكروه على من ستسمع      ومن بعد ما أبدي يسن و يشرع

مصل و تال ذاكر و محدث

Allamah Shami explains the above by saying:

قوله سلامك مكروه : ظاهره التحريم ط

(شامي 1:455)

Meaning it is highly reprehensible to say salam to such people. He further states:

ياثم بالسلام على المشغولين بالخطبة او الصلوة او قراءة القران

(شامي 1:456)

It is a sin to say salam to the one who is busy in salah, khutbah and the recitation of the Quran. He further writes that if someone says salams to these people then it is not wajib on them to reply to the salam.

 يكره السلام على المصلي والقارئ ، والجالس للقضاء أو البحث في الفقه أو التخلي ولو سلم عليهم لا يجب عليهم الرد لأنه في غير محله . ا هـ . ومفاده أن كل محل لا يشرع فيه السلام لا يجب رده

(شامي 1:457)

What did you mean by the statement in the note? An answer can only be given if it is explained.

And Allah Ta’aalaa knows best

(Hazrat Mufti) Ahmed Khanpuri (May Allah preserve him)

The answer is correct

(Mufti) Abbas Dawud Bismillah (May Allah preserve him)

(Mahmoodul Fataawaa 4/712-713)