Producing Oil well at BHP below BP can lead to
1- Evolution of dissolved gas in and near wellbore (into the reservoir) which in turn effect relative permeability of oil to flow in the pores and make oil thicker (increases viscosity). Lower relative perm and higher viscosity translates into lower mobility of oil and thus production rates at surface.
2- In case you have produced the oil below its BP and then you attempt to initiate water injection, keep in mind that gas is also present in reservoir with oil and/or connate water so sweep efficiency of water flood will be reduced owing to presence of three phases in the porous medium. Also, thicker oil will be difficult to be swept by water.