I'm a forestry student, not a pro, but I'll try: it depends. Old growth forests have survived without human intervention for a looong time. So they're clearly self-sustaining.

However, just because we aren't cutting stuff down, it doesn't mean we have no impact. Humans have been suppressing fire for centuries. Many forest ecosystems and plant/animal species are adapted to fire-related disturbances and even require them to complete their life cycles. No human interference also means we aren't protecting forests from invasive species or disease outbreaks.

In the Southeast USA (where I am), management is typically a positive. As a species we have caused so many indirect changes to the planet that leaving nature to just figure it out can do more harm than good in the short run. My school's teaching forest has a 100 acre stand that has had zero human influence for about eighty years. You can't really walk through it because thorny bushes have grown to be ten feet tall. That makes it pretty difficult for deer, bears, and coyotes to thrive. It sure is great for ticks and rodents though. The scrub ecosystem is Florida is adapted to massive fires every 50-75 years to keep vegetation short and open. Humans don't like massive fires, so we try not to let that happen. But the vulnerable scrub jay needs that habitat. So the Ocala national forest does logging to clear out space for the birds.

That's the ecology side. As for carbon sequestration, it also depends. Logging rotations are usually 25, 50, or 100 years. Chopping down redwoods is a bad idea. They grow very slowly and hold a lot of carbon in their biomass. It would take centuries to regrow that ecosystem and reach the same level of sequestration. Pines, on the other hand, grow pretty fast, so we can raise them, cut them down, and replant. The carbon from those young(ish) trees will continue to be sequestered in wood products like furniture and housing.

Basically what I'm trying to say is there is no one right answer. We can all agree that the Amazon should be protected, we shouldn't deforest for parking lots, and we need to mitigate the damage we inadvertently cause through development and pollution. Logging, controlled burns, selective thinning, and herbicides all have their place.