Praise be to Allah.
Yes, there are du’aa’s that are narrated in the Sunnah which the fasting person may say at the time of breaking the fast. He may say “Dhahaba al-zama’u wa abtalat al-‘urooq wa thabat al-ajru in sha Allaah (Thirst has gone, the veins are moist, and the reward is assured, if Allaah wills).” He may also say whatever du’aa’ he wants, not because that is mentioned specifically in the Sunnah, but because it is the end of an act of worship, and it is prescribed for the Muslim to say du’aa’ in that case.
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allaah have mercy on him) was asked:
Is there any du’aa’ narrated from the Prophet (peace and blessings of Allaah be upon him) which is to be said at the time of breaking the fast? When should it be said? Should the fasting person repeat the adhaan after the muezzin or continue breaking his fast?
The time of breaking the fast is a time when du’aa’s are answered, because it is the end of an act of worship, and because a person is usually at his weakest when breaking his fast. The greater a person’s sense of helplessness and weakness is, the closer he is to humility before Allaah. The du’aa’ that is narrated is: “Allaahumma laka sumtu wa ‘ala rizqika aftartu (O Allaah, for You have I fasted and by Your provision I have broken my fast).” And the Prophet (peace and blessings of Allaah be upon him) said: “Dhahaba al-zama’ wa abtalat al-‘urooq wa thabat al-ajr in sha Allaah (Thirst has gone, the veins are moist, and the reward is assured, if Allaah wills).” Although there may be some weakness in these two hadeeths, some scholars classed them as hasan. Whatever the case, if you says these du’aa’s or some others when breaking the fast, this is a time when du’aa’s are answered. End quote.
Majmoo’ Fataawa al-Shaykh Ibn ‘Uthaymeen (19/question no. 341)
For information on the isnaad of these hadeeths (Dhahaba al-zama’ and Allaahumma laka sumtu) please see the answer to question no. 26879, where it states that the former is hasan and the latter is da'eef, and refers to a fatwa of Shaykh al-Islam Ibn Taymiyah concerning fabricated du’aa’s.
With regard to sahoor, there is no specific du’aa’ to be said at this time. What is prescribed is to say Bismillah at the beginning and to praise Him when one stops eating, as should be done at every meal.
But the one who delays his sahoor until the last third of the night also catches up with the time of the divine descent, and this is a time when du’aa’s are answered.
It was narrated from Abu Hurayrah (may Allaah be pleased with him) that the Messenger of Allaah (peace and blessings of Allaah be upon him) said: “Our Lord, may He be blessed and exalted, comes down to the lowest heaven every night when the last third of the night is left, and He says: ‘Who will call upon Me, that I may answer him? Who will ask of Me, that I may give him? Who will ask Me for forgiveness, that I may forgive him?’” Narrated by al-Bukhaari (1094) and Muslim (758). So he should say du’aa’ at this time because it is a time when du’aa’ is answered, not because of sahoor.
With regard to the intention, the site thereof is the heart, and it is not prescribed to utter it out loud. Shaykh al-Islam Ibn Taymiyah said: “Whoever thinks in his heart that he will fast tomorrow has formed the intention.”
And Allaah knows best.