How many different due dates are there? If many cases share the same due date, you could economise by using the multi-case edit feature.
However, I agree that your suggestion sounds like a nice improvement.
based on the comment that you provided to my original response, I would suggest that perhaps you should consider exactly how you are using FogBugz, and whether it is the right tool for you. The "due date" feature was originally designed as an aid for making sure that emails entering FogBugz received a timely reply. It then became a handy feature for project managers to apply pressure to developers. Anyone can get a report from FBz as to which cases are behind their due date, allowing PMs to send "reminders".
The "due date" feature was not intended to be a way of replicating a GANTT chart in FogBugz.
My suggestion is to identify key delivery milestones, and any "internal synchronisation points" within your project. Put them in as "Milestones" on the project (you need admin level permission on the project to do this), and give them "Official Due Dates". FogBugz will assume that the temporal order of the Milestones must be maintained ---- this is the way that you get automatic dependencies in FogBugz (explicit dependencies between milestones is also supported, so that you can have dependencies between projects.) Try to resist the urge to give every case its own unique milestone, because that will be even more painful than having to update the due date for each and every case. If it seems that every case must have its own milestone, then consider whether your cases are at the right level of granularity. If all of your answers are suggesting that you need a GANTT chart tool, then go and get MS Project.
If you choose to follow this advice, then the only step left is to assign the cases to the new Milestones, and do away with the due-dates.