I wanted to create a Tasker task to open my computer's browser to the same url that my phone's browser is on. I can do this by copying the url and running a task to share it, but that's too much work. After much digging, I found a way to extract the current url from the browser database.
sqlite3 /data/data/com.android.browser/databases/browser.db "SELECT * FROM bookmarks;" | tr '|' '\n' | grep http | tail -n1
If you go to cnn.com on the phone and run that command, the output is http://m.cnn.com/.
Using that, I can make a short script to make it open on my home computer via ssh:
#! /system/bin/sh
url=$(sqlite3 /data/data/com.android.browser/databases/browser.db "SELECT * FROM bookmarks;" | tr '|' '\n' | grep http | tail -n1)
ssh -i /PATH/TO/SSHKEY USER@IP "firefox $url"
If I'm browsing a page on the phone and want to read it on the computer instead, I can swipe Wave Launcher and select the task that runs this command. This will grab the last url visited, so it doesn't matter if the browser is open or closed.