In most cases No. Unless the water has a very low amount of
     dissolved O2 in it, or is really mucky which results in a lower O2
     level.  The betta can do both, breath with their gills like other fish,
     and Using the labyrinth organ to process atomospheric air.  The surface
     should be open at least a little bit.  But in general good clean water
     which is changed at least once a week will provide all the needed O2
     that a betta needs.