Trees, yes. Forrests, maybe not how you are thinking. Theyare referred to as "stochastic". Trees fall, other species move in for a bit, old tree type returns. Fire wipes out a patch, grasses, followed by bushes reclaim, but the Trees come back. Over a long enough time scale, because of Tectonic motion, the Trees may end up at a latitude they are not well adapted to. The forests that once grew on the land thar is now Antarctica certainly got well and truly screwed.

They wax and wain. They move, they change.

Even where I grew up, the woods are always changing. Some of our woods are old growth that have been changing over as the old White Pines die from blister rust. That opens up the canopy and regen comes up in it's place. Usually Aspen and Alder. In the meadow, where no White Pines have sprouted in hundreds of years, suddenly they start to spring up like magic when they feel the old tree has died. They compete until one major tree grows and the others whither. The Red Pines are expanding into the meadow to the east. When I was a kid, it was almost completely open. Now it's almost 100% covered with Red Pine. The twisted blue cedar remains the unchanged in it's little enclave. Unchanging. Next to the cedar swamp is the black spruce bog. All muskeg and skinny trees that could be 400 years old for all we know. Once you're inside the bog, it's pitch black at high noon. The ground is untrustworthy, you should stay out in the summer.

The creek meanders and new oxbows form. Where an oxbow lake appears, the Balm of Gilead comes up along with Diamond Willow. The Bam grows faster than weed. Many feet per year. Eventually, the oxbow dries up, the creek keeps moving, the beavers cut down all the Bam and aspen and make a new dam. Now there's a new pond for a while killing off some of the trees which will attract creatures that will live in the dead tree stump.

Tomorrow, new things will happen.

Ooh, so there's a lot to discuss on this topic! So back in the mid twentieth century, when the science of ecology was just getting started, there was a popular theory that forests were essentially indefinite without disturbance. After a disturbance like a tornado or forest fire, ecological succession would occur and end in a stable 'climax community' of tall canopy trees, understory trees, and finally forest floor vegetation, and that the climax community was essentially static. The terms super-organism and old-growth forests were coined to describe these systems, and many ecologists considered these ancient forests to be in perfect balance, where every single species had an important role to play, no species could be removed from that system without a fundamental breakdown, and once that balanced climax forest was created, it would remain forever until some new disturbance arrived. (Here is one of the most popular early papers on the topic: Clements, F.E., 1936. Nature and structure of the climax. Journal of ecology, 24(1), pp.252-284. - just google scholar it, the pdf is free if you want to read it)

However, we don't really think of forests this way anymore. Forests are essentially just ecosystems that are defined by the presence of large trees. If you have large trees, you have a forest, and large trees are remarkably competitive in areas with the climate to support their growth and where disturbance is infrequent or not severe enough to kill them. So if there is relatively little disturbance and suitable climate, some kind of forest will tend to remain for as long as conditions allow it to. However, the specifics within each forest will change dramatically in response to thousands of factors: presence of pathogens and herbivores, climate, moisture levels, man-made and natural disturbance, and much more.

Probably the biggest factor in determining the 'longevity' of a forest is disturbance, which varies depending on conditions. In the eastern United States where conditions are humid and moist, the main disturbance factor (exculding man made ones like logging or deforestation) is actually small scale wind throw events. In those forests, as trees get really old and big, they eventually become so heavy and catch so much wind that a particularly bad storm will uproot them, knocking them over. They often then fall into other trees so you get a 'domino effect' of one to a few dozen trees falling over in the forest. That creates holes in the canopy which allows in more light and a small area of a different community type that is often closer to a grassland or shrubland than a forest. However, this doesn't last long as new trees grow tall and surrounding branches close in to cover up the hole in the canopy. There has been plenty of research on this type of disturbance, but here is one example looking at the extent of these disturbance events (Ulanova, N.G., 2000. The effects of windthrow on forests at different spatial scales: a review. Forest ecology and management, 135(1-3), pp.155-167.)

However, in fire-mediated forests, like those of the pacific northwest, fire is the main disturbance factor. Everyone's heard about the giant, terrible forest fires that have been rampaging through the region in recent years. Those are serious, forest-ending disturbance events (at least for a few dozen/hundred years until the canopy can re-establish itself), but they are not the norm. Normally, every inch of these forests will burn once every 5-10 years (a period known as the fire-return interval), and because the fires are so frequent, all they do is burn off the underbrush and take out any old/rotting/weak trees - an effect not unlike the gap phase disturbance model of eastern forests. The only reason we have so many massive firestorms nowadays is because of smoky the bear - human efforts to put out any and all forest fires as soon as they start created a giant buildup of fuel and dead stuff on the forest floor for decades, so now when the forests inevitable do catch fire, all the fuel burns like crazy and creates flames so big that they destroy healthy canopy trees as well. Humans meddled in the natural nutrient cycling regimes of those ecosystems, and now we're paying the price for it. (Steel, Z.L., Safford, H.D. and Viers, J.H., 2015. The fire frequency‐severity relationship and the legacy of fire suppression in California forests. Ecosphere, 6(1), pp.1-23.) I should note here that this does not occur in Eastern forests because the main mechanism for cycling dead plant matter in these systems is decomposition, not fire. The environment is generally so moist and humid that fungi can break down even gigantic dead trees in a matter of just a few years, so the fuel does not build up as plants die. Furthermore, the fire return interval for most eastern forests is around 500 years, 100x less than for fire-mediated ecosystems.

So to get back to your core question - the answer is complicated. If you're asking whether forests just naturally 'peter out' due to lack of resources or some such over time, the answer is generally no. In a healthy forest ecosystem, all nutrients are cycled back into the system somehow, be it from fire or decomposition, and even though individual trees may die and species regimes may shift over time, some sort of forest canopy will almost always remain as long as climate and disturbance regimes allow it to.

But as we've seen with recent wildfires, human intervention in natural forest ecosystem processes can cause huge cascading effects that can wipe out entire forests if we're not careful. Whether those human destroyed forests will return is something that remains to be seen. Now that there are so many invasive species, we realistically cannot expect the same trajectory of ecological succession that we've seen in the past, so the species composition of our regrowth forests will be different, but it is likely that some form of forest will eventually regrow in those areas.

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Yeah they have life cycles.

The stage at the beginning, or rebirth, of a forest is known as an early successional stage. Essentially the area starts with plants such as moss, shrubs etc. This land over time transitions into a forest (depending on its location). Natural disturbances can destroy forests but over time it will go through the stages and eventually rebound.

There is debate on whether forests have a “climax stage.” In other words, a stage at which the forest reaches where changes no longer occur. I do not know too much about that debate so I’ll link an article when I find it.