Can destruction be profitable? We often see news that some company destroys items they couldn't sell in time.

Also we can see news of supermarkets destroying food just because it's not fresh although it's perfectly edible. 

If someone destroy goods, he would get $0 for it or even need pay for the disposal. How can this act be profitable for a company?
It can be profitable for the monopolist to do so. For the conventional producer who is a price taker the profit objective function looks like this:
$$\max_{q} \Pi^c $$
where $\Pi^c = P \cdot q - C(q)$.

That is, they seek to maximize profits, facing an exogenous price to sell goods and where costs are a function of amount produced. If everything is nice and differentiable and concave this gives a first order maximizing condition of:
$$ 0 = \frac{\partial\Pi^c}{\partial q} = P - \frac{\partial C(q)}{\partial q} \rightarrow q = g(P) $$  where $g(P)$ is the inverse of function $\frac{\partial C(q)}{\partial q}$ (increase q from zero until marginal cost equals the price).

But for a monopolist $\Pi^m = P(q) \cdot q - C(q)$. That is, the monopolist is not a price taker, they know that when they produce more prices will fall. Resulting FOC:

$$ 0 = \frac{\partial\Pi^m}{\partial q} = P(q)+q(\frac{\partial P(q)}{\partial q}) - \frac{\partial C(q)}{\partial q}$$

As a result, a monopolist in this setting will not want to sell where price equals marginal cost but rather a higher price than marginal cost:
 $$ \frac{\partial C(q)}{\partial q} - q(\frac{\partial P(q)}{\partial q})= P(q) $$
Recall: $- q(\frac{\partial P(q)}{\partial q}) &gt; 0$

Indeed, it is this selling of goods above their marginal cost that is the source of the monopolist's monopoly rents. 

Now back to the question itself which asks about when "company destroys items they couldn't sell in time." We can think of inventory as a good with low or zero marginal cost, so yes, by the argument above, it can be profit maximizing to not sell goods with a price above costs as long as it lowers the profits on the other goods you sell enough to offset the additional revenue. 

Other forces are likely at play as well. Stores often worry about their brand value and low quality, low price, goods may harm their relationship with their customers. Many people shop at stores specifically for the high quality goods and others at different stores for low prices and it may be too costly to move spoiling goods from one venue to the other and cheaper just to toss it. We can also think of the muffin stumps of the TV show Seinfeld as an example of this high transaction costs / missing markets problem.  

That said, there are sometimes innovative alternative methods of disposal:


  Rotisserie chickens have been around for a while. I used to bypass
  them and roast my own, until I noticed something: The rotisserie
  chickens were actually cheaper than buying and roasting my own.
  
  Cat Vasko noticed the same thing and decided to figure out why. The
  answer makes a surprising amount of sense: Grocery stores make them
  out of unsold chicken that is about to pass its expiration date. It's
  an elegant way to make a profit out of food that would otherwise be a
  net loss. And it's not just chicken -- according to Vasko, the
  ever-expanding prepared-foods section of the supermarket uses up all
  sorts of unsold produce and meat. It is, as she says, a bit like
  hunter-gatherers using every inch of the animal.


Everyone Wins When You Buy a Rotisserie Chicken