Do I need to add support to hold second floor joists while working?
Do you need too?
Probably, but that may depend on exactly how much is above the header/load bearing wall you're taking out.
Should you?
Yes. Build a temporary wall to support the joists, ceiling, upper floors, whatever is above this. The cost in time and supplies is minuscule in comparison to any damage that may happen to the structure while you've got that load unsupported. Additionally, your insurance company could use the lack of temporary bracing as an excuse to not pay a claim, making it even more expensive.
While you're at it, ensure that your load is carried all the way down to the ground. You don't indicate if you've got a slab, basement, or crawl space, but if you're not on a slab, you'll want to build a temporary wall in the basement/crawl space directly below the one you're building above. This will support the joists right where the load is instead of bowing them down with the weight from above. Again, consider the actual costs vs the potential costs to repair damage.