Would wearing a sling help relive the pain from bursitis? .

Somewhat.: Assuming this is bursitis in the shoulder, a sling might help ease the pain when you're standing/walking, but I doubt it would help when sitting or lying down. Also, using a sling too much can contribute to "frozen shoulder" (AKA "adhesive capsulitis") so it's important to move the shoulder joint in as full a range of motion as tolerable regularly to prevent this condition.