Praise be to Allah.
It is proven in the saheeh hadiths that the devils are chained up during the month of Ramadan.
It was narrated that Abu Hurayrah (may Allah be pleased with him) said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “When the month of Ramadan begins, the gates of heaven are opened, the gates of Hell are closed, and the devils are chained up.”
Narrated by al-Bukhaari (1899) and Muslim (1079).
But this chaining up does not mean that one should not seek refuge with Allah from the accursed Shaytaan during the day in Ramadan, especially in cases where it is prescribed to seek refuge with Allah from him, as is the case when starting to read Qur’an or when entering the washroom and the like. This is for two reasons:
The first reason that is the hadith states that the devils are chained up and that this continues throughout the month of Ramadan, but it does not state that their whispers (waswasah) stop at all.
Abu’l-Waleed al-Baaji (may Allah have mercy on him) said:
The phrase “and the devils are chained up” may be understood primarily as meaning that they are indeed chained up and prevented from doing some things that they can only do if they are free. That does not indicate that they stop doing things altogether, because the one who is chained up has his hand tied to his neck, but he can still speak, see and move around.
End quote from al-Muntaqa (2/75).
For more information on what is meant by the devils being chained up, please see questions no. 39736 and 12653.
The second thing is that seeking refuge with Allah from the accursed Shaytaan is prescribed and enjoined in several cases, and includes seeking refuge with Allah from the evil suggestions and whispers (waswasah) of the Shaytaan when he starts to whisper. Allah, may He be exalted, says (interpretation of the meaning):
“And if an evil suggestion comes to you from Satan, then seek refuge in Allah. Indeed, He is Hearing and Knowing”
Similarly, it is prescribed to seek refuge with Allah from the Shaytaan when wanting to read Qur’an. Allah, may He be exalted, also says (interpretation of the meaning):
“So when you recite the Qur'an, [first] seek refuge in Allah from Satan, the expelled [from His mercy]”
This means that seeking refuge with Allah from the accursed Shaytaan is a matter of worship that is prescribed, so it is not valid to say that there is no point in doing that at any particular time, unless there is a text from the One who prescribed that in the first place, because this is a matter of the unseen and reason plays no role here. As the Lawgiver has not exempted Ramadan from the general meaning of the command to seek refuge with Allah from the Shaytaan, there are no grounds for making such an exception simply on the basis of rational reasoning, even if we accept that the devils are chained up in Ramadan, because all of that has to do with what the Lawgiver has told us and enjoined, and there is no contradiction between the two [namely, the command to seek refuge with Allah from the Shaytaan and the fact that the devils are chained up during Ramadan].
To sum up: the Muslim should continue to seek refuge with Allah from the accursed Shaytaan in the cases where it is prescribed to do so, and he should not stop doing that just because of an idea that crosses his mind or a specious argument.
And Allah knows best.