I made a simple map with polygons of the staates of africa and used the rule-based renderer for different colors to tint them. When I tried to load or export this map with qqis2web I didn't get displayed any colors but only the contours. I tried several very simple rules but in every case the result was always the same. If I used the categorized renderer for that everything worked fine. Isn't the qqis2web able to display rule- based rendering or what could be the matter in that case?
It all depends on your rules. Can you post (some of) them? The rule-based renderer is tied heavily to QGIS expressions, and only a small subset of QGIS expression functions are so far supported in qgis2web. Some missing functions will be easy to add, others less so, and some pretty much impossible.
So, if you can edit your question to put in some or all of the expressions used, I can give a more definitive answer.