As a leader, it can be tempting to push your team to get the most out of them and to get results. People don't like like saying no and often just want to be helpful and give it their all. Many people will even offer to do things like take a meeting on a personal day or work a bit later for a week or two. While it's important to respect people's privacy and their right to manage their own lives, it's also your responsibility to do what you can to discourage burnout.
I’ve found the best ways to manage this is to limit the amount of unnecessary stress at work, keep your ear very close to the ground, and use smoke signals. Some strategies
Lead by example, I once saw a manager leave a note in slack saying they were going to take that day off because they "woke up grumpy". That's awesome leadership, it encourages others to do the same and normalizes the idea of a mental health day.
Experiment with mandatory time off; my last role had flexible PTO so it was originally a safeguard to prevent people from not taking it. Apart from a minimum, we said that part of it must be consecutive. I made it clear that I would deduct from bonuses if people don't take enough.
Be clear and transparent. Don't make people stress out about unknowns. While constant reprioritization is a stressor, once you know about something it doesn't hurt to fill the team in. If you are setting expectations define them as much as possible and make yourself around to answer questions if/when there ends up still being unclarity.
Don't write or respond to emails after work hours; I may be working but I don't want to publicize (and certainly not glorify) it or compel people to respond. Gmail has a feature to schedule emails for the next day, so does slack.
Respect people’s privacy. Let people know they don't need to give a reason for taking time off and make sure not to ask for details unless people volunteer them.
Be there if someone is working off-hours. In my line of work, it was very rare for anyone to have to work nights or weekends. For the few times we needed to I made sure to be there (in person or virtually) as well.
Smoke signals; besides traditional 1-1s and retros, I did a weekly session where I take a bunch of gifs and put them on a miro board, then everyone goes around and talks about which one they feel most like. I had a "dumb manager" idea and created a dashboard to emulate a Brazilian steakhouse, (put up a green card if you need more stuff to do, put on a red if you are full), no one really used it but I don't regret trying.
At the end of the day, people will live their own lives. Stress can come from many places and while you can never eliminate it completely, it doesn't hurt to try to reduce it if you can. A less stressful team is more effective, happier, less likely to leave and just results in a more fun environment.