I do not know for certain why you are getting this error but I do have 2 suggestions that sometimes work for me:
1. Download the dashboard and open locally in Desktop, let it connect to the server or wherever it does NOW. go to the Data Menu, scroll down to the source and select create local copy from the submenu. That will create a local copy of the dataset. Sometimes Tableau offers you the choice of using your local copy, if not the next step will help.
2. Even if you can't do that, try this with your Schema -
A. Open the dashboard in desktop. (after downloading a local copy if you haven't already.)
B. Select Data from the Menu and then New Data Source.
C. Then go ahead and select your new database.
D. Next create a new worksheet and pull 1 field from the new datasource onto the page. I usually do the row shelf. E, Save and then go to an existing worksheet that uses the other datasource. (e sure it is a worksheet, not a dashboard)
F. Use Data/ Scroll to the datasource. Select Replace Datasource from the menu.
My ideas, questions, things to check as to why from my own painful experiences:
Is the current dashboard connecting live or using a published datasource?
If a published datasource are there calculated fields that are not in the new source ?
Are the character sets the same between the 2 ? (usually that will throw an error that tells you in my experience).
If you are connecting to the new source via the server, does the server have rights to the source.
Are the two sources the same version of the database engine? SqlServer 2012, 2014 and 2008 do not always "play nice" . The same can occur with service packs in the same version. I have been out of the arena of other databases recently, but similar things have been known to happen.
Karen, it seems your steps work well. This morning I've used them successfully. Thanks for the instructions. This has been very helpful! You're my hero!!