It's about resource, it is not about money
Having burned out at least three times for both paid and unpaid projects (not programming related, but this is a human question, not a technical one) I think I can definitely offer a perspective on this
For the most part burnout occurs because you're given too much to do for your capabilities, available energy, and insufficient recovery time
Some people will therefore be more able to cope than others, and for different types of projects
Burn out is not linear, so if burn out is starting there needs to be an immediate, substantial, and ongoing reduction in demand
Once burnout has actually occurred, helping is largely a case of bolting the stable door. At that point you need to immediately stop working on the project. You may never be able to go back.
There are things that can help, unfortunately a lot of them don't match well to open source :
Gather more volunteers. For commercial companies, employ more. Spread load and enforce time off to prevent burnout
Thank contributors. In companies, an above inflation pay rise is also a fundamental necessity to know you're being appreciated.
Actively include contributors in the associated community/company. Manage any criticism that comes their way.
If criticism is necessary, be constructive, and offer assistance.
If someone specifically says they are burning out, listen to them, but if this hasn't been noticed previously someone isn't doing their job.
Be realistic in your requirements.
As a sub note on criticism and requirements : don't cause drama either through human interaction, or making changes that generate substantial work, without consideration and communication first.
If someone has to step away due to burn out, do not criticise them for doing so, even if this means higher pressure on other people.
There's a problem here, because support roles that exist in companies may not exist to the same degree (or at all) in open source projects. People are also less likely to volunteer for roles which have no visible end product.
I've seen a lot of people burn out, and in difference to the stupid guff at the beginning of these comments it spread across the spectrum of social class, gender, age, sexuality, and race.
For non paid projects, it's worth considering if there is a natural life span for volunteering. Few people carry on volunteering forever, some areas have an average time people contribute before moving on. Actually a lot of that applies to paid jobs too!
If you're not receiving sufficient community involvement/enhancement/thanks to your life for contributing it is only a matter of time before burn out occurs. Plan your retirement and step away when you can.
Volunteers are human yet users of non commercial offerings often have unrealistic expectations. This cannot be overstated. I've repeatedly seen small teams burn out from organising events due to a lack of resource and extensive requirements. Unless expectations are managed or volunteers increase, this will eventually destroy a community.