Is it permissible to pray or fast based on the adhan of prayer apps on the phone, or must one follow the local muezzin despite the difference in his adhan's timing and its occasional inaccuracy?

The times for the five daily prayers are tied to perceptible signs, not merely to the adhan, as the Prophet, peace and blessings be upon him, clarified. The adhan is for announcing the entry of the time, but the prayer is not valid unless one is certain or has a strong presumption that the time has entered. The muezzin must observe this, and worshippers must not pray when in doubt or when they know that the adhan was called before the time. This also applies to the adhan broadcast from electronic devices. As for fasting, its matter is easier, as one can abstain (from food and drink) before its time enters.