Welcome to the community! I think this is a great question that many people might be wondering about. Probably the most dangerous for the liver is ethanol being absorbed or inhaled from the handwash (the majority of a hand sanitizer is alcohol).
I found a study looking at this: https://www.sciencedirect.com/science/article/pii/S0735675706001318?via%3Dihub. It was quite small (only 5 people) but they got them to apply “5 mL of the product (62% denatured ethyl alcohol manufactured by Kimberley-Clark, Roswell, GA) to both hands and rubbed until dry. This activity was repeated 50 times over 4 hours.” This is probably a lot more than what you are doing in your day to day (even with COVID). Long story short, they could not detect any increase in ethanol levels in the blood after this study.
Others have looked at isopropanol hand sanitizers (getting participants to use it every 10 minutes!), finding a very small increase in the blood. https://www.journalofhospitalinfection.com/article/S0195-6701(04)00013-1/fulltext
These suggest that the sanitisers aren’t doing anything to your liver health. We can also rest easy a bit with data from health care workers, who use huge amounts of hand cleaning products but do not see any real increase in liver disease.
Hope this helps,