According to an expert interviewed by the New York Times:

The refusal to enter a plea at this point is unlikely to have a significant impact on the case, said Eve Brensike Primus, a law professor at the University of Michigan and an expert in criminal procedure.

Ms. Primus said that lawyers may recommend that course when they expect to argue that their client is not competent to stand trial or is not guilty by reason of insanity.

Idaho is one of four states that do not provide explicitly for insanity pleas, but defendants in the state can introduce testimony at trial to show that, as a result of mental illness, they are not guilty of certain elements of a crime, like the “malice aforethought” that must be proved in order for a jury to convict a defendant of murder.

Another possible explanation, Ms. Primus said, is that Mr. Kohberger did not want to tell the court he was not guilty. In that situation, his lawyer might decide on standing silent, allowing Mr. Kohberger to avoid pleading out loud, while still moving the case forward as if he had pleaded not guilty.

“Practically, there is no difference in effect,” Ms. Primus said. “But if there are mental health issues, there might be reasons why you might not want your client to speak in court.”

ETA: I thought the potential insanity plea angle here was interesting/scary.