The resolution required two changes:

Update the manifest to remove the section indicating that it requires
administrator access.

Update the app.config to include the supportedRuntimes attribute.

It was crashing with .net 1.1 except when run as administrator, so I had
updated the manifest to run as administrator.  However, the app will run with
.net 2, so I had to update the app.config to indicate that it supports 2.0 in
order for it to choose .net 2.0 at startup instead of 1.1.  As soon as it
could start without elevated priviliges, the drag and drop started working as
well.