AppleTV 3 on a LAN network (no internet) with a PPC Mac and iPhone
The iPhone and the AppleTV should work just fine together (with or without internet).   Note that some router models do not work well together with AirPlay so if you encounter transmission problems this should be the first suspect.

Regarding your Mac I believe that the latest iTunes for OS X 10.5 can do AirPlay. If you run an older version of OS X you should consider the most appropriate upgrade path.  It might give a better experience to buy a new mac instead of upgrading to OS X 10.5