The obligatory "HR is not your friend" comment has already been dropped in the comments. But that doesn't automatically equate to "never tell HR anything".
It is possible that the colleague is aware of their situation and knows that the "suffering at work" treatment isn't the solution. The source of their current stress could be of a personal nature, and unrelated to their work. Maybe you just notice that he is struggling, but are wrongly attributing it to work as opposed to other things you do not know about.
It's not the colleague's responsibility to inform you of any personal issues they may or may not be having.
On the other hand, people on the verge of burnouts are not the best judges of themselves (I can speak from personal experience here). First of all, they are likely to continue the behavior that drove them on the verge of a burnout since it's what got them there in the first place; and secondly, the impending burnout may be affecting their self-awareness even further.
There is an argument to be made to talk to HR, not just for the colleague's benefit, but also the company's benefit. If this person's work cannot be done by others, and they're on the verge of a burnout, that's going to cause problems for the company too. Addressing this before they actually burn out is a win-win-win for the colleague, the company, and any concerned coworkers.
Let's look at it another way. Let's say this colleague is needlessly performing a dangerous task, e.g. not wearing their helmet on a construction site. It could cause harm to them. On top of that, as an essential worker the company/project would suffer from their absence as they cannot be easily replaced. But the colleague tells you to let him be. Would you?
Whether it's mental harm or physical harm, the principle of the thing remains the same.
Personally, I would talk to management in an unofficial capacity. Not to officially start the "suffering at work" procedure, but rather to address a concern that you are not sure of. I suggest to specifically stress that you're not sure whether there is a problem or not, since that is the actual truth.
If the company then decides to start the "suffering at work" procedure on their own, that means that they could've also done this by themselves if they had been the ones to notice the coworker's current state of mind and singlehandedly concluded that this is an impending burnout.
In other words, you're only sharing your observations and genuine concerns with the company. It's the company who decides to start the procedure.
To what extent, and under which conditions should a suffering colleague be helped against his will?
If it is conclusively provable that he is in fact burning out due to his work, then there is a moral argument to act in the best interest of the coworker - as far as you're legally allowed.
But mental distress is usually not provable unless investigated/treated, so this may be an impossible option for this particular scenario.
If it's an unproven concern (no matter if the group all shares that concern or not), then you should at most discuss your concern rather than their problem. Don't run to the manager claiming that there is a problem.
Simply share your observations and ask the manager if they've observed similar things, and that you (as a group) are willing to help out. Let the company take it from there.
It seems to me like you're in the latter category here.
Regardless of whether you address this with the company or not, I agree with Phillip's suggestion about working at increasing the bus factor. It may help the colleague, but even if it doesn't it's still something that should be done purely for the company's (and employees') best interest.
