A statute is a rule, regulation, or law.
A statute of limitations imposes a limit on how long after a crime is committed a person can be charged with that crime. This phrase is used almost exclusively in a legal context, unless it's being deployed metaphorically or humorously for effect.
The examples below illustrate this clearly:
Ms. Gievers suggested that we file individual lawsuits on my behalf soon because of statute of limitations laws. (Three Little Words)
In February, the judge dismissed the lawsuit, saying it was barred by a three-year statute of limitations. (Washington Times)
A statue is a sculpture. The word looks and sounds a lot like statute, but its meaning is completely unrelated. If you made a clay model of a speed limit sign, that might qualify as a statue of limitations, but otherwise the phrase is meaningless. It's an eggcorn, an incorrect expression that arose out of mishearing a word — like hearing eggcorn instead of acorn. Nonetheless, as with so many similar misunderstandings, it finds its way into print far too often:
But Manhattan District judge Alison Nathan said Charles had taken too long to sue, as the statue of limitations was three years. (BBC)
But few corruption trials make it to court in Argentina, and those that do often drag out so long that the statue of limitations comes into effect before sentencing. (The Guardian)
Statue of limitations is just plain silly, unless you're a sculptor with a fondness for eggcorns. Statues belong in museums, not in law books!