Is there any example of living beings destroying their environment?
Species don't live in isolation
As simple as the question seems, it actually contains some issue in the conceptualization of a species in isolation that affect or does not affect its environment.
Species do not exist in isolation from others. Many species have very important impact on their environment. These transformations would yield the environment to be unsuitable for the survival of the species over time if there were no other species to interact with. But by the existence of other species creating a cycle of, say oxygen for example, the transformation that the species is bringing to their environment is not seen as destroying the environment. In fact, if they were to disappear that could be an important problem. Let's go through examples
Example 1: Cyanobacteria and the Great Oxygenation Event
In the history of our planet, about 2.45 billions years ago, some species started to produce oxygen. Before that the oxygen level on earth was very low. We refer to this rise in oxygen as the Great Oxygenation Event. At the time, there was very few species that were performing respiration (breaking up oxygen and producing carbonic gas). Oxygen is a very reactive molecule and the rise in oxygen level was actually a MASSIVE environmental impact (arguably the most important life-induced environmental that had ever happen on Earth). It lead to an event of mass extinction.
The main contributors to this rise in oxygen were cyanobacteria. Cyanobacteria were most definitely destroying their environment. However, today, while modern cyanobaceteria are still producing loads of oxygen, we don't consider they are threatening the environment. In fact, they are fixing quite a bit of carbonic gas, which we rather tend to see as beneficial.
Example 2: Ants
I will not develop this example too much... Ants are greatly affecting forests of the world. Without ants, forests would be pretty different (see this non-peer reviewed source of information).
Ants are typically considered indicator of a &quot;healthy&quot; forest (see Segat et al. 2017; Lawes et al. 2017 and the book Gorb and Gorb 2003). Howe ver ants can also be invasive species that cause massive &quot;destruction&quot; on their habitat (Tsutsui and Suarez 2003; Lach 2003; Ness and Bronstein 2004).
&quot;Destruction&quot; is context specific
So, wether a species is &quot;destroying&quot; it's environment depends upon what other species are around. It also depends upon what you are aiming for (species diversity vs stable communities vs ...) but I won't go into these details.
Instead of asking

What species destroy their environment

it would not be 'more wrong' to ask

What species are present in ecosystem that are not adapted to their presence

or even (although that is a little of a stretch and some may argue on it)

What species have recently been introduced in a new ecosystem? / What species have recently changed their environmental impacts?

Always keep in mind, that no species can live sustainably on its own alone.
Do humans destroy their environment?
What do we mean when we say that humans destroy the environment, then? Human activities have changed drastically very recently. This is caused for two main reasons 1) our technology and, consequently our needs and therefore consumption have changed drastically ever since the industrial era. 2) we went through a demographic explosion.

So, yes we are pumping lots of carbon dioxide in the atmosphere. We get this carbon from forests we cut down, from large carbon reserves that have accumulated over millions of years (petrol, gas, peat and others). This change in our environmental impact is often seen (and for rather good reasons) as destructive.
Other modern species that destroy their own environment
Pretty much any invasive species would qualify as an example of a species that destroys its environment in a way that is not sustainable for itself over the long term. Among the invasive species that have important environmental impact that are seen as highly destructive, I can think of

Emerald ash borer (North America)
European starling (North America)
Feral hogs (North America)
Brown tree snake (North America)
grey squirrel (North America)
Zebra mussel (Europe and North America)
Japanese knotweed (Europe)
Cane toad (Australia and Latin Amercia)
Northern Pacific seastar (Australia)
Common wasp (Australia)
Rabbits (Australia and New Zealand)
Crown-of-thorns starfish (Indo-pacific and Australia)
Water hyacinth (global)
black wattle (global)
Argentine ant (global)

Many of these examples are from western countries because 1) that's what we tend to have most data on 2) because that's what I know having only lived in Europe, North America (and a little bit in Indonesia) 3) There are many examples from Australia and New Zealand because they are the world champions in invasive species and in bad policies concerning these invasive species!
Some readers will probably complain (and some with good reasons) about the species listed as what we mean by &quot;destroy&quot; is very subjective and my list is not intended to show the species that are considered &quot;most destructive&quot;. Also, the locations of where these species are invasive might be inaccurate and I welcome anyone to edit them.
You can also consider any predator (incl. herbivores) and parasitic species that is leading it's prey / host to extinction such as

The experimental case with lizards in the carribean (Schoener et al. 2001)
The atlantic salmons leading shrimps to extinction (with the help of human activities) in the baltic sea
The red foxes affecting (with the help of human activities) hares population in Europe.

Coming back to the question

Is there any example of living beings destroying their environment?

Yes... other species than humans are &quot;destroying&quot; their environment. Estimating how much they are destroying their environment is really hard to both define and measure. Without a good working definition and without looking at actual data, I would personally be happy to consider humans as being the species that &quot;destroys&quot; the most today.