For Rails 4, see this answer.
For Rails 3.x, configure a logger in config/environments/test.rb:
config.logger = Logger.new(STDOUT)
config.logger.level = Logger::ERROR
This will interleave any errors that are logged during testing to STDOUT. You may wish to route the output to STDERR or use a different log level instead.
Sending these messages to both the console and a log file requires something more robust than Ruby's built-in Logger class. The logging gem will do what you want. Add it to your Gemfile, then set up two appenders in config/environments/test.rb:
logger = Logging.logger['test']
logger.add_appenders(
Logging.appenders.stdout,
Logging.appenders.file('example.log')
)
logger.level = :info
config.logger = logger