Maybe here's an example that's more useful: Some years ago, I worked on an in-house IaaS platform that included several workflow modules for the sysadmins (alert dispatching, time-tracking, etc.).

One of those components was a scheduler for one-time and recurring tasks. For example, when you delete a system, you set a timer for 14 days to have the system remind you (via e-mail or by issuing an alert into the queue) to delete the system's backup. There were also a lot of recurring tasks that needed to be performed daily or weekly. Now if you have a task thats configured daily at 9 AM, it's tempting to implement the timestamp series as

  while (true) { timestamp += 86400; }

And indeed, that's how it was done in the existing code. But that means that once DST starts or ends, your daily-at-9-AM-task suddenly happens at 8 AM or 10 AM instead. Whether that's a problem depends on the type of task and how the sysadmins organize their work. And then there's the monthly recurrence, which is even messier with plain timestamps.

I cannot recall all details anymore, but I definitely remember that twice a year, after each DST change, someone would go through the list of recurring tasks, and move the times forward (or backward) by one hour manually.

EDIT: Maybe the simplest (though not easiest) solution to the irregular month lengths would be to attach giant thrusters to Earth and push it away from the sun a bit, so that our year is 366 days instead of 365 long. Then we make a calendar with 6 months per 61 days. As a bonus, it would reverse some of the effects of global warming. (Alternatively, go to 368 days and have 16 months with 23 days each.)

It depends on the application semantics, for example if I scheduled a Breakfast for 08:00 in Las Vegas, but then moved the event to Chicago, I wouldn't want the time to be auto-converted to 06:00, (or whatever is appropriate given DST), because that's a silly time to have breakfast.

It doesn't need to come up "a lot" for your application to feel buggy. It just needs to come up.

Your UI has a selector for Date/Time and a selector for Location. They might even be different screens. The Location may have been prefilled with a guess based on geoip. Or maybe they saved the wrong 'Springfield'.

The user took an action to change Location. If that has the side effect of changing Time, the user will be surprised.