If you ignore papers notifying you that a lawsuit has been filed against you, things could go very wrong. If that happens, filing bankruptcy might fix it for you.
If you ignore papers notifying you that a lawsuit has been filed against you, the person or company who sued you will likely get a "default" (read "easy") judgment against you.
That means the court would declare that you owed them so much money. Note that you would have bypassed your opportunity to defend against the lawsuit, possibly even by showing that that collecting the debt was barred by the statute of limitations.
Large buyers of debt sue people all the time for time-barred debt, but if no defense is made, the court issues a judgment for the debt buyer. Then the debt buyer can take money from your bank account or garnish your wages. If this happens to you, contact an experienced bankruptcy attorney in your area ASAP. It's possible that you might even be able to get back the money taken if you act fast enough.