As a word in its own right borough is pronounced ˈbʌrə, but as a place-name or personal name element it is usually pronounced bərə or, even more commonly, brə. Here are some examples:
Farnborough Gainsborough Middlesborough
The same pronunciation is also used for the place-name element burgh, as in:
Aldeburgh = ˈɔːlbrə
Edinburgh = ˈedɪnbrə