The ticking noise represents the technology and the impact it has had on the house as time passes. 

Something terrible happened in this story that turned the people into nothing but spots in paint.  There was a catastrophic event, probably a nuclear bomb.  Since it was obviously a technology that caused the damage, created by people, the house’s technology is significant.  Throughout the story, the house ticks itself closer to its doom. 

In the living room the voice-clock sang, Tick-tock, seven o'clock, time to get up, time to get up, seven o'clock! as if it were afraid that nobody would. The morning house lay empty. The clock ticked on, repeating and repeating its sounds into the emptiness. 

The house is fully automated, and these automations are designed for the convenience of the people.  It is comforting people who are no longer there.  However, they also make the house almost alive too.  It continues to cater to people who are long gone, until its own inevitable destruction. 

The ticking of the clock not only represents the technology that has destroyed the people, it also foreshadows their doom.  As the clock ticks its way through its routine, it is getting closer and closer to the fire. 

Eight-one, tick-tock, eight-one o'clock, off to school, off to work, run, run, eight-one! But no doors slammed, no carpets took the soft tread of rubber heels. It was raining outside. The weather box on the front door sang quietly: "Rain, rain, go away; rubbers, raincoats for today…" And the rain tapped on the empty house, echoing. 

The people do not come back.  They are never coming back.  Yet the storm that destroys the house is an act of nature, not technology.  The house cannot save itself.  Slowly but surely, everything is going back to nature, and back to nothing.  The clock will tick no more.