I am posting this here in case others have the same issue. The three errors all fixed by this are:

"Error: Assertion Failed"
"Error: Illegal Argument"
"Error invoking service"

All in Plan view specifically in Developers Taskboard mode and only in the web client. The Eclipse client operates without errors.

I saw all three of those errors for the same problem in different browsers. In this case "Illegal Argument" is the correct error. It is caused by something in an ordered list being an invalid selection. For example, if you had a task with a Priority of 5 and then deleted 5 as an option for Priority in enumerations, the selection on that task for Priority is now invalid and will generate an error in the web UI even though it does not in Eclipse. Thanks to this article for the tip:

https://www-304.ibm.com/support/docview.wss?uid=swg21391241

However, that's not what caused the error for us. In our case the error was caused by an invalid State. I had put the "Deferred" state in the group of "None". Despite the system letting me do that, the web UI considers the state to then be invalid even though Eclipse is ok with it. A little bit of a bug, but nonetheless I fixed the issue by putting the Deferred state back into the group "Open". Which then re-triggered the problem of items getting marked as Deferred erroneously, so then I fixed that too. I'll post that solution here:

https://jazz.net/forums/viewtopic.php?t=18037

More fascinating trivia. I viewed several sprint plans on the web in Dev Taskboard view to see which ones were broken. Four out of 13 sprint plans and the Backlog all generated an error from the above list of error messages. Fixing this problem made the error go away for all of those, even though only sprint 13 and the Backlog had deferred items! I originally speculated it was related to parent/child relationships, but that was not the case. Curiouser and curiouser. My next speculation is its related to where the ID numbers of the Deferred items fall, but since the problem is now fixed there is no value in my tracking this down further. Just know that even though the plan you are trying to view appears broken, the work item breaking it may not actually be in that plan.