Certainly not! Not in every case, that is.
It's funny how many people think that "natural" means almost
the same thing as "safe", as though they'd never heard of
poison ivy or deadly mushrooms.
Some natural dyes are almost perfectly safe; others are quite toxic. Some synthetic dyes are safe even to eat; others are too toxic to bring into your home.
Logwood, for example, is a lovely natural dye, capable of creating a surprising range of colors, depending on the mordant used, including violets, blue-greys, and the best natural black. However, the active ingredients, hematein and hematoxylin, are toxic whether inhaled, absorbed through the skin, or ingested. As much care should be taken with this natural dye as with any of the common synthetic dyes we use. Unless a given dye, natural or not, has been tested for safety or is commonly used for food, it is important to follow these same precautions. Wear gloves, don't breathe the powdered form, and never reuse a dyeing pot for food preparation!
Most natural dyes are not particularly toxic in themselves,
but they will not stick to fabric unless a mordant is
used. Typically, a mordant is a heavy metal. Heavy metals are
extremely toxic, and bad for the environment, as well.
Alum is a relatively safe mordant that is often used with natural dyes. It
is any of several chemical compounds based on aluminum,
which (in spite of the silly claims about Alzheimers's
disease, which is not caused by aluminum) is much safer than
chromium and other heavy metals.
A somewhat less safe mordant is iron. Iron is toxic
in overdose (according to the United States FDA, iron is the leading
cause of poisoning deaths in children under 6, despite
child-resistant packaging), but it is, of course,
nutritionally desirable in tiny quantities, and it will not harm the environment when disposed of. Iron is not only
a mordant for other dyes, but can be used as an interesting
dye in its own right, by applying bits of iron metal to
fabric and allowing them to rust. At least it does not poison the ground if you pour this mordant out on it.
(Most natural dyes work much better on wool than on cotton,
so wool dyers will generally be more enthusiastic about
natural dyes than cotton dyers.)
When using natural dyes, be careful to research the safety
of your mordants!