Q: Is it permissible, even after finding out that a hadith is da’eef (weak), to follow it, by way of doing righteous deeds? The hadith says: “When it is halfway through Shaban, spend that night in prayer and fast on that day.” The fast is observed as a voluntary act of devotion to Allah, as is spending that night in prayer (qiyaam al-layl).

Firstly: What is narrated concerning the virtue of praying, fasting and worshipping on the fifteenth of Shaban (al-nusf min Shaban) does not come under the heading of da’eef (weak), rather it comes under the heading of mawdoo’ (fabricated) and baatil (false). So it is not permissible to follow it or to act upon it, whether that is in doing righteous deeds or otherwise.
A number of scholars ruled that the reports concerning that were false, such as Ibn al-Jawzi in his book al-Mawdoo’aat, 2/440-445; Ibn Qayyim al-Jawziyyah in al-Manaar al-Muneef, no. 174- 177; Abu Shaamah al-Shaafa’i in al-Baa’ith ‘ala Inkaar al-Bida’ wa’l-Hawaadith, 124-137; al-‘Iraaqi in Takhreej Ihyaa’ ‘Uloom il-Deen, no. 582. Shaykh al-Islam [Ibn Taymiyah] narrated that there was consensus on the fact that they are false, in Majmoo’ al-Fataawa, 28/138.
Shaykh Ibn Baaz (may Allah have mercy on him) said in Ruling on celebrating the fifteenth of Sha’baan:
Celebrating the night of the fifteenth of Shaban by praying etc, or singling out this day for fasting, is a reprehensible bid’ah (innovation) according to the majority of scholars, and there is no basis for this in Shariah.
And he (may Allaah have mercy on him) said: 
There is no saheeh hadith concerning the night of the fifteenth of Sha’baan (Laylat al-Nusf min Sha’baan). All the hadiths that have been narrated concerning that are mawdoo’ (fabricated) and da’eef (weak), and have no basis. There is nothing special about this night, and no recitation of Quran or prayer, whether alone or in congregation, is specified for this night. What some of the scholars have said about it being special is a weak opinion. It is not permissible to single it out for any special actions. This is the correct view. And Allah is the Source of strength.
Secondly: Even if we assume that the hadeeth is da’eef (weak) and not mawdoo’ (fabricated), the correct scholarly view is that weak ahaadeeth should not be followed at all, even if they speak of righteous deeds or of targheeb and tarheeb (promises and warnings). The saheeh reports are sufficient and the Muslim has no need to follow the da’eef reports. Nothing is known in Islam to suggest that this night or day is special, either from the Prophet (peace and blessings of Allaah be upon him) or from his companions.
The scholar Ahmad Shaakir said: There is no difference between rulings or righteous deeds; we do not take any of them from da’eef reports, rather no one hast he right to use any report as evidence unless it is proven to be soundly narrated from the Messenger of Allaah (peace and blessings of Allaah be upon him) in a saheeh or hasan hadeeth.