Subqueries return more than 1 row

VJ Reddy's solution will get you rid of the error message. But the problem is different.
With your UPDATE query, you are about to update ALL rows of table1, with one value. I.e. all rows of table1 will have that facilityID value from table2 which was retrieved as the first value from the subquery. I am pretty sure that that is not intended, is it?
This means that you are missing a WHERE clause in the UPDATE query proper. Could you describe in plain English which further conditions must be applied to get the "correct" facilityID to the "correct" row in table1?

You can also try this code:

Update table1 set facilityID =
(select a.facilityID FROM table2 a, table3 b
WHERE a.name =b.PROV_ORG_NAME LIMIT 0,1)

Well, also your entire query needed to be rethink-ed. i can guess you might needed to make a function to solve the issue.

As Bernhard Hiller pointed out this will update all of your rows. If you're trying to fetch the correct facilityId for each row you should have a correlation in the sub query to the row that's being updated. For example something like (without knowing your actual need or table structure)

Update table1 set facilityID = (select facilityID 
                                FROM table2 
                                where table2.name = table1.prov_org_name)