Using the MicroPython logging module and coding examples, along with a little help from the Python logging handlers source code, I was able to create a MicroPython custom handler class for sending log messages to a network-based syslog server. You can find it on my GitHub page:

https://github.com/DavesCodeMusings/SyslogHandler-MicroPython

In a nutshell, you need a network attached MicroPython device and the IP address and port of a syslog server on your network. Using MicroPython's logging.getLogger().addHandler() method, you create another destination for your log messages. They'll still print to the console, just like they would with a regular logging setup, but you'll also get a copy sent to your syslog server.

For example, on the console:

INFO:root:Test message Info
ERROR:root:Test message Error
WARNING:root:Test message Warning

And in the syslog server's log file:

Aug  2 02:23:15 192.168.1.102 Test message Info
Aug  2 02:23:15 192.168.1.102 Test message Error
Aug  2 02:23:15 192.168.1.102 Test message Warning

You can get it installed with mip.

mpremote connect PORT mip install logging
mpremote connect PORT mip install github:DavesCodeMusings/SyslogHandler-MicroPython

A sample main.py shows how to get it set up from there.

Have fun logging everything on the planet!