Is it permissible to interrupt and disrupt the Friday sermon, and what are the legitimate Sharia justifications for that?

Speaking while the Imam delivers the Friday sermon is forbidden, and the one who does so is sinful, even if their speech is remembrance of Allah, due to it being idle talk and a violation of the command to listen attentively.

Evidence: On the authority of Abu Hurayrah, the Prophet, peace and blessings be upon him, said: "If you say to your companion while the Imam is delivering the sermon on Friday, 'Listen attentively,' you have engaged in idle talk." On the authority of Abu Darda, he said: "Ubayy spoke the truth: If you hear your Imam speaking, listen attentively until he finishes." The prohibition includes responding to a sneeze (tashmeet al-'aatis) and returning greetings, as well as remembrances that are uttered aloud. Saying "Ameen" to the preacher's supplication and sending blessings upon the Prophet should be done silently.

The prohibition of speaking and engaging in remembrance applies when the preacher is delivering the sermon from the minbar. However, if the preacher is present but not delivering a sermon, there is no prohibition on speaking or remembering Allah. The hadith (definition) "When the preacher ascends the minbar, there is no prayer and no speech" is baseless.

The following are exceptions to the prohibition: Unavoidable necessities such as drowsiness or answering the call of nature. What is permissible during prayer, such as guiding a blind person or necessities that prevent destruction or the loss of a great benefit (like the prayer for rain - du'a al-istisqa'). Correcting an error of the preacher or prompting him. Responding to the preacher if he advocates polytheism, innovation, or evil, provided it does not lead to a greater mischief. The prayer of greeting the mosque (salat tahiyyat al-masjid). Signaling to someone who is speaking to be silent without uttering a word.