You should never assign 127.0.0.1 to eth0 -- that is for loopback (and should be the IP of the lo interface).
To change it you can do:
ifconfig eth0 <new IP> netmask <netmask> up
Note that you might have to set your gateway and DNS servers (via the route command and editing /etc/resolv.conf, respectively).
This change will not persist accross reboots. Is your tinycore installation installed to your hard drive, or are you booting off CD/USB?