Can I die from serum sickness if I ignore it?

Yes. Serum sickness can lead to organ damage and vascular collapse if the offending agent is not stopped. Normally however it is a self-limited syndrome which resolves after the offending agent is removed.
Rare but possible. The greatest danger is kidney injury resulting in acute renal failure. This is more likely if you already have compromised kidneys from pre existing conditions. You might need dialysis for a few weeks until your kidneys recover. The key in serum sickness is monitoring - adding therapy when needed until the problem resolves.