How do the townspeople treat the lottery box?

The box is an interesting part of this story, and the town's treatment of it is equally interesting.  I would say that in general, the people treat the black box with a mixture of wary respect and carelessness at the same time.  
Let us start with wary respect.  Essentially, the black box determines who lives and who will die.  It is not a nice box that is holding the name of a traditional lottery winner.  The "winner" of this lottery is stoned to death, so the box represents a scary set of final moments for a single person every year.  This lottery has been going on for years and years and is an important part of the society.  The box is the central item around which the lottery takes place; therefore, it is a powerful and scary item.  People whisper in its presence, and they make sure to maintain a certain distance from it.  

When he arrived in the square, carrying the black wooden box, there was a murmur of conversation among the villagers, and he waved and called. "Little late today, folks." The postmaster, Mr. Graves, followed him, carrying a three-legged stool, and the stool was put in the center of the square and Mr. Summers set the black box down on it. The villagers kept their distance, leaving a space between themselves and the stool.

In addition to being wary of the box's power, the people are unwilling to get rid of it.  It is not some meaningless box that can be easily replaced.  The box, like the lottery itself, is part of a deep tradition.  It must be respected as a part of that tradition.  That is why nobody supports the idea of getting a new box.  

Mr. Summers spoke frequently to the villagers about making a new box, but no one liked to upset even as much tradition as was represented by the black box.

Despite all of the above, it is shocking how carelessly the people treat the box on any other day of the year.  While the box has a special stool that it sits on during the lottery, it has no such special resting place for the other 364 days of the year.  It is stored wherever a space can be found, and there is nothing special about a place like the back of a barn.    

The rest of the year, the box was put way, sometimes one place, sometimes another; it had spent one year in Mr. Graves's barn and another year underfoot in the post office. Sometimes it was set on a shelf in the Martin grocery and left there. 

This general carelessness is probably why the box has become so shabby looking.  People do not take care of it, and it has become a beat-up, old box. 

The black box grew shabbier each year: by now it was no longer completely black but splintered badly along one side to show the original wood color and in some places faded or stained.

I suppose it is possible that the box's caretakers are intentionally not taking care of the box.  Perhaps they want to be rid of the lottery, and they believe that if the box falls apart, the lottery will end as well.