I've installed OpenWrt onto my La Fonera by telnetting to RedBoot and loading the firmware from my TFTP server (which I already had on my network for PXE purposes), using the following commands:

RedBoot> fis init
RedBoot> load -r -v -b 0x80040450 openwrt-atheros-2.6-root.jffs2-64k
RedBoot> fis create -b 0x80040450 -f 0xA8030000 -l 0x00700000 -e 0x00000000 rootfs
RedBoot> load -r -v -b %{FREEMEMLO} openwrt-atheros-2.6-vmlinux.lzma
RedBoot> fis create -r 0x80041000 -e 0x80041000 vmlinux.bin.l7

Each command executed with no problems.

However...I can't make any sort of network connection to my router.  At this point, I chose to disconnect the La Fonera from my network and connect it directly to my desktop machine, after having installed some tools on it that would help me diagnose the problem.  The IP address of my desktop machine is 192.168.1.1, and the firewall stays disabled, since I generally rely on the gateway to do filtering for me.  The RedBoot default IP address is 192.168.1.40, so now I'll go through my method of figuring out what could possibly be wrong.

I first installed and configured a DHCP server on the desktop, with the following (overly simple) configuration:

ddns-update-style ad-hoc;
subnet 192.168.1.0 netmask 255.255.255.0{
  range 192.168.1.2 192.168.1.253;
  option broadcast-address 192.168.1.255;
}

I ran the dhcpd with the -d switch, since this is for testing, it will display log entries directly to my console, and I don't usually run dhcpd on the desktop.  Since RedBoot is not configured to run a boot script yet, I plug the router into power, login to redboot, and type the following commands:

RedBoot> fis load -l vmlinux.bin.l7
Image loaded from 0x80041000-0x80289086
RedBoot> exec

At this point, RedBoot becomes inaccessible, but the telnet connection to RedBoot does not time out.
A few seconds later, I end up with this in my dhcpd terminal:

Listening on BPF/dc0/00:20:78:12:44:f5/192.168.1/24
Sending on   BPF/dc0/00:20:78:12:44:f5/192.168.1/24
Sending on   Socket/fallback/fallback-net
DHCPDISCOVER from 00:18:84:27:aa:88 via dc0
DHCPOFFER on 192.168.1.253 to 00:18:84:27:aa:88 via dc0
DHCPREQUEST for 192.168.1.253 (192.168.1.1) from 00:18:84:27:aa:88 via dc0
DHCPACK on 192.168.1.253 to 00:18:84:27:aa:88 via dc0

This indicated that the system booted, and obtained an IP address of 192.168.1.253 via dhcp.
At this point, I cannot access the router over the network.  Telnet displays this:

Trying 192.168.1.253...
telnet: connect to address 192.168.1.253: Host is down
telnet: Unable to connect to remote host

nmap displays this:

Starting Nmap 4.20 ( http://insecure.org ) at 2007-03-28 15:10 EDT
Nmap finished: 1 IP address (0 hosts up) scanned in 0.213 seconds

And at this point, I am completely out of ideas, hence this post.  Thanks to everyone in advance for your help.