Hi Andrew,
The reference inputs are high impedance inputs, so you should not see any loading. There should not be a voltage drop when you connect them. Part of the excerise of going through the other steps was to isolate the problem. As the problem appears to be directly related to the reference inputs, all I can think of that could have happened is at some point the reference inputs became damaged. If the ESD cells conducted due to an accidental excessive current flowing through them from exceeding one of the Absolute Maximum Ratings specifications (listed on page 2 of the datasheet) then there may be some sort of short in the ESD cell.
At this point I would suggest replacing the ADS1248 with a new one.
Best regards,
Bob B