The first thing you can do to green a log cabin is make it more energy efficient. For that, you can refer to my previous article here on GreenHomeGuide.
The challenge is that log cabins have an aesthetic look that is readily destroyed by standard energy efficient, nontoxic and dust fighting measures. So focusing on environmentally conscious interior surface finishes, solar and lighting strategies, and warm sweaters are the way to go.
Indoor air quality
Let's look at indoor air quality for a moment. If your log cabin has logs exposed on the interior, it has many excellent dust gathering spots. This could increase the dust mite population and your chance of respiratory aggravation. You could cover the logs up, but then why have a log cabin?
Maybe the solution here is a great HEPA filtered HVAC system or radiant flooring and a built-in vacuum system.
Caulking and sealing
Log cabins require sealants that are the least likely to be made with nontoxic materials. (Clear sealants are harder to reduce VOCs in than pigmented.)
Using a borate based fungicide and sealant is probably the least toxic. Use a joint sealer without mineral spirits. Safecoat may have something recommended for log cabins. With any compound, be sure you are using the correct type for your application, exposure and wood species. Log cabins normally require caulking and sealing, but finding these with low VOC counts can be challenging.
If you add windows on the South side you can warm things up in the winter. Daylighting is a strategy that works in any building, and log cabins are notoriously poorly lit.
Being environmentally conscious about what materials you use in any house is important. If you are trying to keep the character of a wood home and are thinking of using salvaged wood, just be sure you are not importing termites or wood borers.
Manufacturer's sustainability practices
If you are buying your cabin new, you can also investigate the manufacturer's sustainability practices. How are they grading their logs, using their waste and managing their forests?
Some log cabin builders actually use forest floor pick up logs instead of felling their own trees. Although this could be a great way to salvage wood, these logs have a much greater risk of containing beetles.