A big musical could cost $1 million a week or more to run.
A small play could cost as little as about $400K a week.
Most shows fall somewhere in between those two extremes.
This, of course, being a separate expense from the capitalization to actually produce the show in the first place (which can range from about 5-8 million for plays and 10-30 million for musicals).
If you pay attention to the weekly grosses threads the costs are sometimes mentioned in there. If you have specific shows you're wondering about you can ask and see if anyone knows.