Is there a way to do what's asked in the topic subject for unrooted and/or rooted phones?
I'd like to read from a system log file instead of constantly listening the network activity for battery saving purposes.
You can use any SSH client to establish a connection to your host and tail the desired file.
This will not help you saving the battery, since the tail command or any other that collects "off-site" information, needs an active connection that uses network, that in turn consumes more battery.
Personally I use ConnectBot by Kenny Root and Jeffrey Sharkey available from Google Play Store for free.
ConnectBot is a powerful open-source Secure Shell (SSH) client. It can manage simultaneous SSH sessions, create secure tunnels, and copy/paste between other applications.
This client allows you to connect to Secure Shell servers that typically run on UNIX-based servers.