If by logs you mean logging information written by the Log class, I would:
- upgrade to 1.6/develop to have full monolog functionality - copy the MongoDBHandler from Monolog, rename it, and modify it to use Fuel's DB class - use \Log::instance()->pushHandler() to add the new DB handler to monolog
If you only want the DB logging, and no file logging anymore, you will have to extend the Log class, and overload _init() so that is uses your DB handler instead of the default StreamHandler.