My general comments about managing a team that is engulfed in firefighting activities:
1) Don't pile on. Not everything can be a "#1 priority". An important part of the job of a manager is to guide the setting of priorities. Determine the priority ranking, and assign tasks according to it. Shield your team from pressure from other parts of the company when it comes to the items that aren't currently assigned.
2) Set proper expectations to upper management. This is part of your "protect your developers" role -- you are the firewall. Accurately and honestly keep upper management informed of what is currently being worked on, what is not, and why.
3) Don't push your team to work faster than is sustainable. Pushing for extra effort is fine when done occasionally, but it's not fine when it becomes standard operating procedure.
4) Keep track of progress. If it's taking longer than it should to put out a given fire, find out why and grease those skids if possible. If "why" is because it's a trickier job than expected, consider temporarily swapping that task out for a different task of equal priority, or reassigning that task to a different dev, and so forth.
5) Be aware that simply adding more people to a task will not necessarily speed up the completion of that task (and will often slow it down).
6) Celebrate all victories. It's very hard on a team when all they are doing is firefighting. Every time a fire is put out, celebrate that with your team. It doesn't have to be a big deal -- it can be just a department-wide email recognizing the victory, an afternoon off, a cake, or even just a short victory dance in front of the rest of the team. The idea is to provide a sense of completion and a job well done, rather than just letting the team do nothing but dwell on the raging infernos that still await them.
1. Run blameless post-mortems to figure out what caused the incident and more importantly what actions would have prevented the incident from occurring in the first place 2. Prioritize doing the corrective action. This can be as simple as checklists for specific procedures or could be the addition of automated tests/static analysis/linting or automating a procedure. Focus on low-lift quick wins here if possible. You should be selling these corrective actions as required maintenance such as changing the oil in your car. 3. Rinse and repeat.
For SAAS products some things that really make a difference are automated test suites, monitoring, and _documented_ procedures
Create a time tracker where anyone doing unplanned work records it as either firefighting or other unplanned non-firefighting work. Whenever doing planning take the history of these times into account.
Assign one person to be the go-to for any and all fires, have them assess the situation bringing in select other team members as necessary. Having everyone respond to every fire with only a few key people actually doing the extinguishing leads to no scheduled work getting done.
Tech-debt. In my experience a lot of fires result from the tech debt of incomplete or poorly designed/implemented features. Schedule a certain number of hours to reduce this and have a groomed backlog of tech debt prioritized when prioritizing regular schedule work.