Unfortunately, there is not a simple 'yes' or 'no' answer. It is not clear yet if disposable paper cups lined with plastics or bio-plastics can be recycled in Australia. Some states, such as ACT and WA indicate they can be placed in the recycling bin, while others, like SA, direct them to the general waste bin.
There is a lot of confusion in this area from the industry and the consumer alike. Planet Ark talks about the issues succinctly here. They are sometimes accepted across New Zealand and Australia in the co-mingled recycling collections. Call your local Council to find out if they will accept paper cups.
The recycling industry is in further turmoil since China's policy to stop taking 24 types of solid waste from across the world in January 2018. Australia used to export 600,000 tonnes of recyclable waste to China annually. For us, this is the perfect time to look at commercial composting as a serious alternative. See our BioPak Compost Service page to arrange composting services in your area.
The APCO (Australian Packaging Covenant Organisation) is currently working on some guidelines at the national level.