This place-name is derived from OE burh-ēġ ‘fort-island’. There are four known examples (including variants), and in three of the cases the fort referred to was unknown. I managed to find these forts (actually, earthworks) on aerial photos. On the basis of these cases, I would challenge the statement at PN Berks:133 that the names more probably mean "island meadow belonging to the borough" than `island with a burh'. Here and below PN means "The place-names of ... ", referring to the volumes of the English Place-name Society.
Laleham Burway TQ 046 683
See PN Su:110. This can be amended, as a burh is in fact visible: a quadrangular enclosure (not right-angled) now covered by a golf course. The north side follows a former channel of a stream.
2. Borough Marsh, Sonning SU 772 780
See PN Berks:133. A quadrangular enclosure, two sides formed by a stream (incorrectly labelled "Thames" by multimap).
3. Burroway, Bampton SP 308 004
See PN Ox:304. An elliptical earthwork between the Thames and another stream. The shape of this does not fit the pattern of the other three.
Burway, Ludlow SO 499 761
Burgeia 1194 mentioned PN Berks:133. It is clear that the river has changed course, and there are many cropmarks resulting from the removal of field boundaries. But a burh here is still visible, partly covered by the field on the right.