Do I need a pressure tank for my irrigation system?
The key is the pressure regulator. Are you hoping to get a specific pressure of water to the irrigation system, with the regulator turning on and off as necessary? For example, might this pump supply a hose bibb that you can turn off? If so, then you will have to have some storage (a pressure tank) in the pressurized system or the pump will rapidly turn on and off as the pressure oscillates up and down.

If instead you can have the pump turn on and stay on for the duration, with the irrigation system set up so that it accepts whatever the pump gives it, then you need neither the regulator nor the pressure tank. You may actually want to disable the pressure regulator, because if the irrigation system stops accepting water (e.g. you turn off that bibb) and the pressure goes high enough to turn the pump off you'll get that rapid on/off/on/off again.

An alternative would be a pressure relief valve that would allow any excess water to dump back into the source tank. If you did have a relief valve, then you could still have a hose bibb, and turning off the bibb while the pump was on would merely waste some electricity.