When the imaam or a member of the
congregation enters the mosque, should they pray Tahiyyat al-Masjid if they have already
prayed Sunnat al-Fajr at home and then come to the mosque to pray Fajr?
If the situation is as you describe, then when they enter the mosque it is prescribed for them to pray Tahiyyat al-Masjid before sitting down. The imaam may order that the Iqaamah be given for Fajr prayer, after which there is no need for Tahiyyat al-Masjid, because praying the fard (obligatory) prayer is sufficient and means that there is no longer any need to do Tahiyyat al-Masjid. The member of the congregation may also forego Tahiyyat al-Masjid if he enters the mosque when the Iqaamah is being given.
And Allaah is the source of strength. May Allaah bless our Prophet Muhammad and his family and companions.