Seems like half of Melbourne has got a new flat screen TV or computer monitor, judging from the number of CRTs left around the streets.
In my view, you shouldn’t just dump them on the nature strip.
Assuming it’s working, ask around to see if a charity somewhere wants it. Or list it on Freecycle so someone who wants one can come and get it.
If as a last resort it goes onto the nature strip (and of course you should attach the remote and any other accessories to it in a bag), and nobody’s taken it within a few days, don’t just leave it to get rain-damaged and litter up the street. Get it taken away — in a lot of council areas (including here in Glen Eira), you can request hard rubbish collection on-demand, for free.