From Littler:
Governors and public health officials across the country have implemented stringent measures to help contain the spread of COVID-19, such as safer at home and face covering mandates. Some jurisdictions also require employers to screen the health of employees, often as they begin a shift. These health screening steps, including temperature checks, have become more common as states reopen their economies.
This post, current as of October 13, 2020 at 9:00 a.m. (CDT), covers statewide laws and orders that require employers to take employees’ temperatures and/or conduct other employee health screening procedures, such as asking employees about any COVID-19-consistent symptoms using a questionnaire or checklist. This chart covers only generally applicable requirements and does not cover the heightened requirements applicable to certain types of employees, such as healthcare workers; public health workers; long-term care, assisted living, and nursing home workers; first responders; and law enforcement.
There are other limitations of the list that they note, but to see where things are up to on a state level, see their chart.