Ian
You don't mention the census specifically, so i assume you may not have access to it. I can find no mention of a Hannah (either a short or dun , or any other name in Birmingham living at the same address as a George Dunn. A Few records of the census have disappeared, however, so that doesn't mean there wasn't one, or they could have been out and not recorded, or the names misstranscribed. There are a lot of George dunns, but what I have found which might be of some interest is the following:
1871
12, court 1 Gt brook St
George Dunn, boot & shoe maker, 21 born Birmingham
Emma Dunn, Steel pen maker, 20(?) , born Leicester
George Dunn, 1, born Birmingham
1881
3 Chapel Place, (294 Garrison lane)
George Dunn, boot riveter, 54(?), born Aston
Sarah Dunn, pearl button carder, 47, born birmingham
George Dunn, rope & twine maker, 15, born birmingham
Mary A Dunn,scholar, 12, born Birmingham
22 New St Aston
George Dunn,Spur leather cutter, 46, Born Birmingham
No 42 , Court 43 Wilton St, Aston
George Dunn, bootmaker, 32 born birmingham
Emma Dunn, 33. born Leicester
Eliza Dunn, 13, born birmingham
George Dunn, 11, born birmingham
Albert Dunn, 10, born birmingham
William H Dunn, 8., born birmingham
Arthur Dunn, 1 month, born Aston
1891
129 High St Aston
George Dunn,Spur leather cutter, 53, Born Birmingham
No 3, back of 30 Garrison lane
George Dunn, shoemaker, 48
Sarah Dunn, 58, Born Birmingham
George Dunn, shoemaker, 25, Born Birmingham
Mary Ann Dunn, pressworker, brassfoundry, 22, Born Birmingham
1901
52 Alma St
George Dunn, Spur leather cutter,66, Born Birmingham
74 &75 New Canal St (Lodging house)
George Dunn , lodger, bootmaker, 37, Born Birmingham
Mike