When entering an onsen, it is generally best to remove jewelry first. Hot spring water is not just warm water: its mineral content varies from one spring to another, and some of those minerals can cause some metals to discolor, tarnish, or become damaged. If a ring or necklace matters to you, there is little reason to expose it to that risk.
That does not mean Japanese bathers are constantly thinking, “Will this particular spring react with this particular metal?” There are simpler reasons for taking jewelry off as well. A necklace can catch on a towel, a clasp can come loose, or a pendant can be dropped. If something disappears down a drain, it can become more than just the owner’s problem. For anything valuable, taking it off beforehand is simply the safer choice.
That is why seeing someone wearing a necklace in a Japanese onsen would be fairly unusual. Still, this is not really the kind of rule that other bathers are likely to police. If someone forgets to remove a single necklace, people are unlikely to get angry or confront them. The general expectation is to take jewelry off, but in the end, it is also about protecting your own belongings and using your own judgment.
Dyed hair and fabrics are treated a little more casually. Someone who has just dyed their hair may avoid bathing because of possible color fading, but once some time has passed, people generally do not worry about their hair color every time they enter an onsen. The same applies to towels and other fabrics. If something is valuable or easily damaged, you may want to be careful, but ordinary towels or fabrics are not something people usually worry much about.
So yes, onsen water can affect metals, colors, and certain materials, but Japanese bathers are not nervous about everything they bring near the water. Jewelry is generally removed. Freshly dyed hair or something you really do not want to damage deserves a little more care. In practice, that is usually the level of caution involved.