Short answer. No. The order denying either the motion for leave to file a demurrer or the demurrer itself is not reviewable by appeal or by certiorari before judgment. The case continues, and the point has to be raised after judgment through the ordinary remedies.
What the law says
The order denying the motion for leave of court to file demurrer to evidence or the demurrer itself shall not be reviewable by appeal or by certiorari before judgment.
Rule 119, Section 23 — Demurrer to evidence. Read the full provision →
The bar is explicit, and it covers both orders
The rule closes off interlocutory review in plain terms. Neither the denial of the motion for leave to file a demurrer, nor the denial of the demurrer itself, may be reviewed by appeal or by certiorari before judgment. Two separate orders are therefore covered, which matters because an accused may be refused at the leave stage and never reach the demurrer at all. In both situations the answer is the same: the case goes on, and the objection to the ruling is preserved rather than litigated immediately.
Why the rule is drawn this way
Criminal trials would be unworkable if every adverse ruling on the sufficiency of evidence could be taken up separately while the trial waited. The demurrer stage sits in the middle of the proceedings, so allowing review there would let a case be suspended at precisely the point where it is closest to resolution. The rule accepts that some denials will later prove to have been wrong, and treats that as a cost worth bearing in exchange for trials that actually finish.
What happens after a denial
It depends entirely on whether leave was obtained. If the demurrer was filed with leave of court and is denied, the accused may adduce evidence in his defense, so the trial simply continues as it otherwise would have. If it was filed without leave, the accused has waived the right to present evidence and the case is submitted for judgment on the basis of the evidence for the prosecution. That is why the leave decision, made under short non-extendible periods, deserves more thought than the demurrer itself.
Preserving the point for later
Because immediate review is unavailable, the practical work lies in the record. The motion for leave is required to state its grounds specifically in any event, so make sure those grounds are set out properly and that the ruling and its basis are clear. The insufficiency of the prosecution's evidence remains available as a matter to raise once judgment has been rendered, through the ordinary remedies, and a well-made record is what keeps that argument alive until then.