That's a really tough situation, especially if you don't know the full root causes of your manager's problems.
How to best to handle it will depend on the sort of relationship you have with your manager.
If your relationship has little-to-no personal dimension then it's probably not something you can really approach directly. That doesn't mean you can't be of some help though - in simple terms you need to try and lighten the load. If you know if tasks they have building up that you can take care of you can offer to deal with it for them, if there are decisions or questions that you can resolve without needing to involve them (and you have the authority/autonomy to do so of course) then do that. As I'm sure you'll remember from your own experience when you are up to your eyes and teetering on the edge of burning out getting even the smallest of things of your plate can help massively. Some quick Do's and Don'ts:
Don't seek out any credit or rewards for doing any of this, otherwise your noble intentions could be easily misconstrued as an attempt to seize on their percieved weakness to further your own goals.
Don't bring the topic of their possible burn out up directly with them (unless they give you a very natural opening) - there may be things going on in their private life that they would much rather keep out of the workplace and if they are in a bad enough state of mind with personal issues that they are in pure white-knuckling-it mode to keep going at work you risk seriously upsetting them or bringing the whole house of cards crashing down.
Do make use of others in the team if you can - try not to bring up the context of the manager's burn out but try and head off problems before they get to him where possible. This applies not just to the decisions and questions I mentioned earlier but is especially valid for the sort of petty, meaningless trivialities that seems to be [i]de rigour[/i] in most offices - "Wah! Jimmy turned down the air con *again" and I'm cold" that sort of thing.
Only you can really know if you are close enough to your manager to do this or not but if you think you are then you could approach them and ask if they are doing ok and if there is anything you can do to help. Unless you know them very well and are reasonably certain as to the reasons why they are feeling this way I'd make it clear that you aren't there to pry, instead you are there to listen and help if they want it. To minimise the potential for any shame/embarrassment they might feel and to maximise the chances of them opening up to you make sure you do this in private and when you are reasonably certain that there can't be any interruptions. End of the day when most people have gone or at the start before people arrive is probably best for that sort of chat.
EDIT: Just seen the OP's latest update saying they are strictly co-workers and haven't known each other long so the "Friends" section doesn't really apply here, I'll leave it in for now though as it may be useful to others but can remove if required.