The tap water at my house seems to have a lot of chlorine, does this pose any health risks to drink?

All Depends-Relative: No way to answer remotely. Likely most of what you see in the water, especially sediment once water has evaporated, is caco3, see: https://en.Wikipedia.Org/wiki/calcium_carbonate (dissolved limestone), not chlorine, see: https://en.Wikipedia.Org/wiki/water_chlorination ; other minerals/solutes. Low dose cl to kill bacteria is typically healthier than not, but h2o analysis required to better know.