Does forgetting to pray the Isha and Tarawih prayers invalidate the fast of the following day, and what should be done about the missed prayer?

Your fast is valid and is not affected by missing the 'Isha and Tarawih prayers. As for the 'Isha prayer you missed due to forgetfulness, there is no sin upon you for it, but you must make it up as soon as you remember, based on the Prophet's (peace and blessings be upon him) saying: "Whoever forgets a prayer, let him pray it when he remembers it." As for the Tarawih prayer, jurists have differed regarding making it up; the Hanafis and Hanbalis believe it is not to be made up, while others consider it recommended to make it up as a supererogatory (nafl) prayer.