Can someone please advise me of a formula that would automatically update the
number of weeks from a list of dates to the current date?

If the dates are in column A, put this in B1 and copy down:
=(TODAY()-A1)/7

This should put the "number of weeks" next to each date in the
original list.

Column B can be formatted as a number with decimals if that's the way
you want to report fractions of a week. Otherwise, you could wrap
ROUND() or INT() or ROUNDUP() around the formula, depending on the
need.