Is the publisher of Islamic topics accountable for the sins of those who browse forums containing forbidden sections, after his topics were published there by visitors and frequenters of Islamic forums? And do you advise him to continue?

There is no impediment to spreading good words that call to goodness and virtue on a website where truth mixes with falsehood, and there is no harm in copying and publishing them on other similar websites. The one who spreads good incurs no sin if their writings lead some to enter mixed websites and be influenced by their falsehood. So, continue spreading good.