Yes!
Where?
How does insulation work?
Foam of Fibre?
Fibres
Foams
Form
Shape and Size
Safety
Strength
Open or Closed?
Moisture Content
Moisture Permeability
Temperature Range
Heat Capacity
Thermal Conductivity or U value
Ecological Decisions
There are many different insulation materials available. They come in different forms and give different performance in terms of heat, airtightness, moisture and sound, and they have different price tags. There is no single "best insulation material". Architects and builders may have favourite materials, and manufactures usually promote their own materials and point out shortcomings in competing materials (for example this online article by the founder and managing partner of Havelock Wool).
There are also different impacts on environment during manufacture and disposal, but to quote Schmidt et al. (2003):
"Many people believe that the emerging insulation products based on biological resources (cellulose), such as flax and paper wool, are much more environmentally friendly than a product based on natural mineral resources such as stone wool. This belief may, however, be unfounded."
If you are worried about fossil-fuel based insulation materials, you need to first consider how much fossil-fuel energy the insulation will save. Sure, you may not be planning on burning gas or oil, but if you're heating with electricity, that electricity has been made with fossil fuels. It may be "green" energy, but the solar panels and wind farms have been made with fossil fuels, and the hydroelectric dams poured with concrete, so you're still not at zero carbon. And even if you did find a way of producing energy without any carbon input, you'd need to calculate whether it's better to use it for your house, or supply that energy somewhere else to offset other people's carbon use.
Schmidt et al. conclude: "The energy and environmental impacts saved during the use phase [of insulation] are more than 100 times larger than the impacts in the rest of the lifecycle".
This sentiment is repeated by the US National Park service for the Pacific Northwest:
"Do not substitute a "green" insulation material for a non-green material if doing so will result in lower overall energy performance. Even though the environmental impacts of the insulation material might be lower for the green product, the overall environmental impact of the building would likely be greater by lower insulating values."
So while important, before you worry about what happened during the manufacture of the insulation, and what will happen if someone burns the insulation at the end of its life, the first thing to worry about is burning less to heat the house while the insulation is doing its job. Building constraints may limit the thickness of insulation, in which case some insulators may not have sufficient performance. Even if thickness is not limited, the extra thickness required for lower performance insulators may require extra materials to keep it in place and more external wall finish to cover it.
It may also be that natural materials attract "natural" pests, molds and fungi. You may want a material that will decompose naturally after its lifetime, but you definitely do not want the material to decompose while it is part of your building. Until the 19th century, the only insulation materials were organic, and the levels of insulation ranged from poor to non-existent. With the new scientific understanding and product availability of the industrial revolution, man-made materials became available, and these were used because they were cheaper, performed better, and would last longer than natural materials. The higher energy costs in the long depression of 1873-1896 boosted use of man-made insulation materials in industrial sites, and a hundred years later the oil shock gave another incentive to insulate homes (see Bozsaky, 2010 for more detailed history). Many "natural" insulators are recent developments that package natural materials in familiar forms of man-made insulation products. They do not necessarily have longer and more reliably determined lifetimes, or lower toxicity than man-made materials, and they may depend on chemical treatment to make them resistant against fire and pests.
In terms of carbon emissions, wood-based products are sometimes considered carbon-negative, since carbon captured by wood is being stored in the building. This may be true, but if the wood were not used in the building would it still be a living tree? Read more about that in a previous post.