Praise be to Allah.
With regard to fasting, it is obligatory to refrain from things that break the fast from the onset of the true dawn, until the sun sets. Allah says (interpretation of the meaning):
“So now have sexual relations with them and seek that which Allaah has ordained for you (offspring), and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night), then complete your Sawm (fast) till the nightfall”
Al-Bukhaari (1919) narrated from ‘Aa’ishah (may Allah be pleased with her) that Bilaal used to give the adhaan at night. The Messenger of Allah (blessings and peace of Allah be upon him) said: “Eat and drink until Ibn Umm Maktoom gives the call to prayer, because he does not give the call to prayer until dawn breaks.”
Based on that, if a person knows that the true dawn has broken, either by seeing it himself or because someone else has told him, then he must stop eating and drinking. If he hears the call to prayer, he must stop eating and drinking as soon as he hears it, if the muezzin give the call on time and not ahead of time.
Some scholars made an exception if a person has the cup in his hand when he hears the adhaan, and said that he may drink what he needs from it, because of the report narrated by Abu Dawood (2350) from Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “When one if you hears the call and the vessel is in his hand, let him not put it down until he has fulfilled his need of it.” Al-Albaani said in Saheeh Abi Dawood: Its isnaad is hasan saheeh. It was also classed as saheeh by al-Haakim, al-Dhahabi and ‘Abd al-Haqq al-Ishbeeli, and Ibn Hazm quoted it as evidence. End quote.
The majority of scholars interpreted it as meaning that the muezzin used to give the call to prayer ahead of time. For a detailed discussion on that, please see the answer to question number 66202.
Most of the muezzins nowadays rely on clocks and timetables, not on actual sighting of the dawn, but this cannot be regarded as certainty that dawn has broken, so if a person is still eating at this time, his fast is valid because he was not certain that dawn had broken. But it is better and more on the safe side to refrain from things that break the fast when one hears the adhaan.
Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him) was asked: What is the shar’i ruling on the fast of one who hears the adhaan of Fajr and continues eating and drinking?
He replied: What the believer must do is refrain from eating and drinking and other things that break the fast as soon as it becomes clear to him that dawn has broken, if the fast is an obligatory fast such as Ramadan or fasting in fulfilment of a vow or as expiation, because Allah says (interpretation of the meaning):
“and eat and drink until the white thread (light) of dawn appears to you distinct from the black thread (darkness of night), then complete your Sawm (fast) till the nightfall”
So if he hears the adhaan and knows that the call is being given at the time of dawn, then he must stop eating and drinking. But if the muezzin gives the adhaan before dawn breaks, he does not have to stop eating and drinking and it is permissible for him to carry on eating and drinking until he sees the dawn.
If he does not know whether the muezzin gives the call to prayer before or after dawn, then it is better and more on the safe side to stop eating and drinking when he hears the call to prayer, but it does not matter if he ate or drank something at the time of the adhaan, because he did not know whether dawn had broken.
It is well known that people living in cities in which there are streetlights cannot see the break of dawn for themselves at the time of dawn. But they should be on the safe side by using the adhaan and timetables which give the time of dawn to the hour and minute, following the words of the Prophet (blessings and peace of Allah be upon him): “Leave that which makes you doubt for that which does not make you doubt” and “Whoever avoids dubious matters will have kept his religious commitment and honour safe.” And Allah is the source of strength. End quote from Fataawa Ramadaan, compiled by Ashraf ‘Abd al-Maqsood, p. 201.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: When should a person stop eating? Is it as they say: when the muezzin says Laa ilaaha ill-Allaah? What is the ruling if he drinks deliberately after the adhaan? Is he like the one who drinks after ‘Asr or can he still fast? Some people say that Fajr is not like a lamp which shines instantly and the matter is broad in scope. What is the ruling?
He replied: If the muezzin gives the call to prayer when the dawn has come, then the Prophet (blessings and peace of Allah be upon him) said: “Eat and drink until Ibn Umm Maktoom gives the call to prayer, because he does not give the call to prayer until dawn breaks.” So if the muezzin says: I have seen the dawn and I do not give the call to prayer until I see the dawn, then it is obligatory for one to stop eating and drinking from the moment he hears the adhaan, except in the case in which a concession is allowed, which is if the vessel is in his hand, in which case he may drink what he needs of it. But if the adhaan is given according to a timetable, then the timetable in fact is not connected to the actual times, rather it is based on calculations -- the timetables that we have now for Umm al-Qura [Makkah] or other cities are based on calculations, because they do not look at the dawn or the sun or the meridian or the time when ‘Asr begins or the setting of the sun. End quote. Al-Liqa’ al-Shahri, 1/214
To sum up: one should stop eating and drinking etc as soon as he hears the adhaan, if he knows that the muezzin gives the call on time. But if he is not sure about that, then he should limit it to drinking what is in his hand, because it cannot be said that he may continue eating and drinking until he is certain that dawn has broken. In fact, he has no means of ascertaining whether dawn has come because of lights and electricity, and many people are unable to distinguish between the true dawn and the false dawn.
And Allah knows best.