Question

Outside of tenure track jobs, what’s the most stressful work environment? Private industry? Government? Something else?

Answer 1

(Assuming we are talking about knowledge workers and excluding applied knowledge workers like doctors in the battlefield for example):
Deadlines, meritocracy, losing and getting jobs, office politics: these are the four main ways that stress comes about in a structural way, leaving aside issues like a bad supervisor/manager/boss.
And it is statistically true that the first three tend to be more rigid/intense/frequent in the private sector than in government. Office politics tend to loom more over your spirit as the organization gets bigger, irrespective of ownership.
And then it has to do with your psychological profile and stage in life of course: some people yearn for meritocracy, others hate it. Some people see the job uncertainty in the private sector as an opportunity, others as a threat. Then after a time, many previously job-hoppers want to settle down.

Answer 2

I'm comfortable ruling out government from the high-stress jobs.  Unless yours is a political appointment in government, the job is likely to be steady and not intrusive in its time demands relative to the private sector or academia.  The downside is that the Peter Principle of promotion is probably most evident in government and the office politics might be very thick, but as the other person answered below, this is not uncommon elsewhere.