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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: $- q(\frac{\partial P(q)}{\partial q}) > 0$
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.
Assuming that the [near] expired goods cannot be sold at full cost anymore, offering them for sale at a significant discount (instead of destroying them) will compete with your own offer of full-priced goods that presumably have much higher margins.
This is pretty much the definition of https://en.wikipedia.org/wiki/Cannibalization_(marketing) which is something that producers have good reason to avoid.
In addition to BKay's answer, selling expired food opens a firm to suit. This occurs either via breach of an implied warranty (that the food is fit for consumption) or, in many jurisdictions, strict tort.
While grocers do carry products liability insurance, the policies require that stores adopt certain standards of practice. It should make sense that refrigeration practices, expiration management, quality control, and the like feature heavily.
This enters a profit function through increasing costs. Specifically, the expected payouts line will increase either by upping the average liability payout (in a simple cash flow approach) or modifying the shape of the distribution of payouts (in a theory context).
Simple answer: It cost companies money to do any, even doing nothing. The immediate issues with selling out of date food in most jurisdictions would be:
Do nothing (stockpile non-perishable.)
Give away or sell at loss:
Back before detailed regulation and stockowner law suits on a hair trigger, officers of companies public and privately-held, large and small, had a little more flexibility in such matters but gradually they have lost almost all discretion. Most regulation today operates under the "everything not allowed is forbidden," model so if the regulators did not anticipate a need for an action, an officer takes the action at his/her personal risk.
Marketwise, there is an interesting analog in the petroleum industry. It's prohibitively expensive to store large amounts of oil, destroying it is economically impossible and today, way to costly in environmental regulations and only a few oil fields have the geology that allows oil to be returned to the ground.
The vast majority of the oil pumped out the ground will be sent through the whole system and end up as an end product 90-120 days at most after it leaves the ground. This lag time causes a significant market inefficiency in petroleum, because "Lifters" those who pump the oil out of the ground can only guess what the demand will be 'X' number of days outward. The oil will be sold period, at some price.
If collectively, the lifters pump to much, the price at the consumer end drop precipitately, but if they pump to little, a spike occurs. When oil prices suddenly plummet, you do see situations where companies are selling oil or oil products at a loss.
Regulation aggravates the problem, particularly the switch over in refining from seasonal versions of products like gasoline as well as regional difference e.g. California, in versions that make the products none fungible between jurisdictions.
The violent oscillations in petroleum prices caused by this lag is one of the reason that petroleum producers from the late-1800s up the present day have such powerful incentives to form cartels and/or have governments regulate the overall oil supply e.g. Texas Railroad commission, OPEC.
It's easy to see that any industry that operated under the conditions that whatever products they had in their channels would end up in consumer's hands eventually, regardless of supply, demand or finale cost, would end up as volatile as the oil industry.
For another example, one could argue that various "agricultural supports" of most nations are primarily devoted to preventing just such volatility. Farmers face the same problem, they plant in spring, harvest in fall, and then sell their crop for whatever the price might be. Perversely, the better the growing season, the more productive the farmers as a whole, the more likely they are to go broke.
In developed nations at least, vastly more amounts of food are destroyed or more often, farmers are paid to never grow it, than get wasted in the distribution system. Most governments believe it's better that food prices remain somewhat higher overall, within certain ranges, than food prices oscillate to much.
