I know this may seem petty, but the VED on a vehicle type is calculated on the emission mass of Carbon Dioxide CO2 produced over 1kM, not Carbon Monoxide CO. CO2 is a gas not a solid.
The VED grammes/CO2/kM rating is produced as part of the vehicles type approval process, and does not change over the life of the approved vehicle.
As Jonny has been stating, the DPF (Diesel Particulate Filter)does what its name implies, it collects particles, such as soot. Soot is fundamentally just Carbon which is a solid. Gasses such as CO2 will pass straight through.
The MOT test looks for the particulates that are emitted from the vehicle, not the gasses. So a DPF should reduce the emission of particulates but again as Jonny has suggested, the limits for the MOT test are set for rather older technologies and any correctly set up modern engine should be well inside the limit even without a DPF present.
The new MOT standard looks to see if a car has been modified. It is unclear to me how that will be applied, and if it would also pick up the removal of a DPF as a modification even if the vehicle still passes the emission tests.
I shall check with my BIL who runs a garage with MOT test station.
If a vehicle has failed an MOT on the basis of a modification then there will be two course of action. The first is to restore the vehicle to manufacturers specification, or the second is to get the modified vehicle reassessed by VOSA/VCA on a single vehicle assessment procedure (like kit cars and single build cars)