I'm always on the lookout for dog friendly places for outdoor adventures with my dogs, I haven't found any overnight walks but I've started some research on the Hume and Hovell track (Albury to Yass), a fair bit of it is on crown land and state forest, dogs are usually allowed in both of these so there should be some sections where you could do a multiday walk with your dog.
Closer to Sydney my favourite dog friendly locations are:
> Red Hill on Sydney's northern beaches - more popular for mountain biking than walking but I used to walk there a lot with my dogs when I lived on the Northern Beaches
> Tahmoor Canyon (near Picton in the Southern Highlands)
> Abrahams Bosom (on the northern headland of Jervis Bay)
> Wingello Forest (near Bundanoon in the southern highlands) - specific mountain bike trails, my younger more athletic dog loves running around the 25km circuit while I ride
for more info checkout
http://www.lpma.nsw.gov.au/about_recreation/walking_trackshttp://www.forestrycorporation.com.au/visiting/activities/hiking-and-bushwalkingPM me if you want more specific info