How do I choose primary keys for star schema BI data warehouse? I am extracting sales turnover data from a Microsoft Dynamics Database into another SQL database that will be used by the BI tool of choice.

This data is coming from G/L (general ledger) entry items.
Each entry contains information for the amount, date, branch, and sales division.

In the DW, I intend to use a star-schema, where the sales figure per day is given in dollars. (fact).

DaySales_Fact

DaySalesId
Sales (Fact)
Date
BranchCode
SalesDiv


Branch_Dim

BranchCode
BranchName


Assume the same thing for Date and SalesDiv tables.

The source data is unreliable and historical data may change at any time. The data warehouse is emptied each day for this reason, and restocked with the live data. This process is scheduled to run once daily.

The source data uses a varchar(2) to store the BranchCode as a primary key.

Would it be best practice to use the business branch code, or create an auto incrementing primary key (see below).

All of the branches are refreshed daily too.

DaySales_Fact

DaySalesId
Sales (Fact)
DateId (fk)
BranchId (fk)
SalesDivId (fk)


Branch_Dim
    BranchId - auto inc PK
    BranchCode 
    BranchName
Almost always surrogate keys (your "Auto Incrementing primary key") are best in data warehouses. I have seen very few exceptions where this is not the case (but some do exist) - yours does not seem to be exceptional. To answer why would be repeating stuff you can find all around the web, by the giants in the field, for instance: http://www.kimballgroup.com/1998/05/surrogate-keys/