Noticed you are looking for Thomas Burrowes. It is possible that the birth could have been registered under the surname Burrows. The BDM website doesn't search under alternative spellings.
Some of the Burrowes/Burrows family appear to buried in the Christchurch City Council cemeteries. http://librarydata.christchurch.org.nz/Cemeteries/
you may need to search under Burrowes and Burrows.