No, judges rarely throw out late claims on their own. Defendants must bring to the court's attention any statute of limitations violation.
To be sure that a judge dismisses an untimely case, you include an "affirmative defense" in your answer, alleging that the plaintiff's complaint is untimely. As an alternative to filing an answer, in some courts you can file a Motion to Dismiss, asking a judge to throw out an untimely complaint.
If you are a defendant who thinks that the plaintiff may have waited too long to sue, you'll need to check the applicable state or federal limitations period to determine whether the lawsuit is timely.
For more information on statutes of limitations, see Everybody's Guide to Small Claims Court, by Ralph Warner (Nolo).