Using black hair dye is a very great sin, whether it is in young age or old age. There are strict warnings in the Hadith for it. There is a Hadith in Sunan Abu Dawood, Pg No. 578:
Translation: It is narrated on the authority of Hadhrat 'Abdullah bin Abbas (May Allah be well pleased with them), that the Holy Prophet (Sallallahu alaihi wa sallam) said: In the last era, there will be some people who will use black hair dye like the chests of the pigeons; they will not get even the fragrance of paradise. (Sunan Abu Dawood, Hadith No. 4214)
It is given in Mishkaat Shareef, Pg No. 382:
Translation: It is narrated on the authority of Hadhrat Jabir (May Allah be well pleased with him) that on the day of the Conquest of Makkah, Hadhrat Abu Quhafa (May Allah be well pleased with him) was brought to the Holy Prophet (Sallallahu alaihi wa sallam). His hair and beard were as white as "Saghama" (a tree with white flowers), the Holy Prophet (Sallallahu alaihi wa sallam) said: Change this whiteness with something else and avoid using black hair dye.
In this Hadith, the prohibition of using black hair dye is clearly mentioned. Thus, we should avoid it and its use is not at all permissible. Apart from black hair dye (or any black color), using any other colored hair dye like red or pure henna is permissible. It is given in Fatawa A'alamgiri, "Kitab Ul Karahiyya":
Translation: It is better to dye the hair and the beard, but "Katam" or leaves of "Neel" should only be used.
In the Hadith, this is clearly mentioned, as given in Sunan Abu Dawood:
Translation: The best dye is henna and "Neel."
However, for a Mujahid, use of black dye is allowed, as mentioned in Radd Ul Muhtaar, Vol. 5, Kitabul Khatari wal Ibaha.
And Allah (Subhanahu Wa Ta’ala) knows best.