Is it permissible to interrupt the Imam during the Friday sermon to ask him to shorten it because of the hot weather?

There is no harm in addressing the Imam to alert him to the intense heat and to not prolong the sermon, provided that it is done gently, respectfully, and courteously. The Sunnah (definition) indicates the permissibility of addressing the Imam during the sermon when there is a need, just as the Companion did who asked the Prophet, peace and blessings be upon him, for rain, and what the Prophet, peace and blessings be upon him, did with Sulayk al-Ghatfani. Speaking with the Imam for a beneficial purpose is not forbidden.