I am doing an experiment ... will the total pressure inside the buret be equal to atmospheric pressure?

These differences may be negligible but are easily measured. Measure the difference in elevation between the meniscus in the burette and the water level; the pressure difference is equal to this number. If you use the mm scale, you will have mm water, then convert to atmosphere or the units of your choice. BTW there are 34 ft of water column to one atmosphere. Remember to measure the barometric pressure because you will not necessarily be starting at 1 atm.