What is the ruling on signing in and out of forums with certain adhkar (remembrances of God), and is this considered a specification of a particular dhikr for a particular time? And does the ruling of expiation for a gathering apply to forums?

There is no impediment to performing dhikr (remembrance of God) when exiting a discussion in a forum. It is legislated when rising from sleep or from any gathering, as evidenced by the Almighty's saying: "And glorify the praises of your Lord when you arise." This is further supported by the hadith of Abdullah ibn Amr ibn al-Aas, may Allah be pleased with him: "Glorified are You, O Allah, and with Your praise, there is no god but You. I seek Your forgiveness and repent to You." Whoever says this in a gathering of dhikr, it will be like a seal, and whoever says it in a frivolous gathering, it will be an atonement for him.