Two of my York Furnace burners ignite and the two closest to flame sensor fail to ignite causing lock out after three tries.

I replaced flame sensor but burners closest to sensor don't ignite so must be another issue.

Any assistance would be appreciated.

Can you remove the tubes to clean them?

There are some kind of crossover tubes or slots between the burners. These tubes or slots are likely dirty. If your burners look something like jet aircraft (have wings), the edges of these wings are the crossovers. Remove the burners & clean out the slots in the wings with a blade type feeler gauge or thin putty knife, then blow them out with air.