In situations like this, we have to understand people didn't really have a lot of materials and stuff we have and didn't do a lot of things we do or can do. I don't think anyone dyed their hair back then in scripture, so you won't find anything like a commandment calling hair dying a sin. People cannot find answers for a lot of things, like video games, processed foods, television, cell phones, masturbation, etc. This is where we need understand why is sin a sin. We can simply take a look at all other verses and things in life that are a sin and why it is obvious when something new in the world comes through, why it would be a sin.
Remember the 80s and such with metal being a new thing and almost every pastor and preacher was going "That music is of the devil!" They were believing only pianos and such were the only godly instruments heh. Not all them but many. Later on, over just a few years we found Christian lyrics and Christians playing that genre with that sound but using it differently from their hearts. Those people were too quick to judge without understanding why that rock or metal would be a sin. It's music in general, it can be used for sin but also for God in another way.
some of the questions you may want to ask sometimes is:
1. does my dyed hair glorify God or make him look bad?
2. Am I doing this to my hair out of hate, revenge, jealousy, envying others, etc?
3. Did I steal this hair dye?
4. Am I using this on my hair to attract men/women to have sex with me before marriage?
Questions like that, and you could ask more, will let you know when your hair dye can become a sin.