Is it prescribed to raise the hands when reciting the adhkaar of morning and evening?

It is not prescribed to raise the hands when reciting the adhkaar of morning and evening, or when reciting adhkaar following the prescribed prayers, because there is no evidence that doing so is mustahabb. 

The basic principle is that it is mustahabb to raise the hands for every du’aa’, except at the times when the Prophet (peace and blessings of Allah be upon him) said du’aa’ but did not raise his hands, such as the du’aa’ during the Jumu’ah khutbah when it is not a prayer for rain (istisqa’), du’aa’ during tawaaf and saa’i, and du’aa’ following prayer, whether what is meant by du’aa’ following prayer is du’aa’ before the salaam or after it. The Prophet (peace and blessings of Allah be upon him) did not raise his hands at these times. 

But if a person raises his hands in some places sometimes, there is nothing wrong with that in sha Allah. 

See the answer to question no. 11543. 

The scholars of the Standing Committee for Issuing Fatwas was asked: Is du’aa’ after obligatory prayer Sunnah? Should the du’aa’ be accompanied by raising of the hands? Is it better to raise the hands with the imam or not?  

They replied: Du’aa’ after the obligatory prayers is not Sunnah if that is done by raising the hands, whether it is done by the imam alone or the worshipper alone or by both of them together. Rather that is bid’ah (innovation), because it is not narrated from the Prophet (peace and blessings of Allah be upon him) or from the Sahaabah (may Allah be pleased with them). As for saying du’aa’ without that, there is nothing wrong with it because there are some ahaadeeth which speak of that. End quote. 

Fatawa al-Lajnah al-Daa’imah (7/103).

Shaykh Ibn Baz (may Allah have mercy on him) said: Raising the hands  in du’aa’ is one of the means of having du’aa’ answered in any place. The Prophet (peace and blessings of Allah be upon him) said: “Your Lord is modest and concealing, and He too shy, if His slave raises his hands to Him for them to come back to him empty.” And he (peace and blessings of Allah be upon him) said: “Allah is Good and only accepts that which is good. Allah commanded the believers to follow the same commandments as He gave to the Messengers. He says (interpretation of the meaning):

‘O (you) Messengers! Eat of the Tayyibaat [all kinds of Halaal (lawful) foods which Allah has made lawful (meat of slaughtered eatable animals, milk products, fats, vegetables, fruits)] and do righteous deeds. Verily, I am Well-Acquainted with what you do’ [al-Mu’minoon 23:51]

‘O you who believe (in the Oneness of Allah — Islamic Monotheism)! Eat of the lawful things that We have provided you with’ [al-Baqarah 2:172]

Then he mentioned a man who has travelled on a long journey and is dishevelled and covered with dust; he stretches forth his hands to the heaven, (saying) “O Lord, O Lord”, but his food is haraam, his drink is haraam, all his nourishment is haraam, so how can he du’aa’ be accepted?” Narrated by Muslim in his Saheeh.

So He has made one of the means of du’aa’ being answered the raising of the hands, and one of the reasons for du’aa’ not being answered is the consumption of haraam food. This indicates that raising the hands is one of the causes of du’aa’ being answered, whether on board a plane or train or in a car or in a space ship or anything else. If a person says du’aa’ and raises his hands, this is one of the causes of du’aa’ being answered, except at the times when the Prophet (peace and blessings of Allah be upon him) did not raise his hands, such as the Jumu’ah khutbah, when he did not raise his hands except when praying for rain, when he did raise his hands. Similarly, between the two prostrations and before the salaam in the last tashahhud, he (peace and blessings of Allah be upon him) did not raise his hands, so we should not raise our hands in these places where the Prophet (peace and blessings of Allah be upon him) did not raise his hands. His doing something is evidence and his not doing something is evidence. Similarly, after saying the salaam at the end of the five daily prayers, when reciting the prescribed adhkaar, he did not raise his hands so we should not raise our hands, following his example. As for the places where he (peace and blessings of Allah be upon him) did raise his hands, the Sunnah is to raise the hands following his example, and because that is one of the means of having du’aa’ answered. The same applies to cases where the Muslim should call upon his Lord in du’aa’ and it is not narrated whether the Prophet (peace and blessings of Allah be upon him) raised his hands or not; in these cases we should raise our hands because of the ahaadeeth which indicate that raising the hands is one of the means of having du’aa’ answered, as stated above.  

End quote from Majmoo’ Fatawa al-Shaykh Ibn Baz (may Allah have mercy on him) (6/158). 

And Allah knows best.