(Scroll to the bottom to get my overall opinion, but as a summary: it could work.)
While Redmine isn't as popular as, say, Bugzilla, they still have a fairly impressive list of 'whose using Redmine'.
The project seems active / healthy. Releases come out every month or two (mostly just minor security updates, it looks like), and it appears as if multiple branches are supported. Support is available from forums and via a running redmine installation; these seem well-utilized, in that there is a lot of comments and changes going on daily. I couldn't find any mailing lists though, which seemed a bit strange. Maybe I just didn't search hard enough.
Cyrus has spoken up with explicit support for redmine. He has firsthand experience and reports that it has no major flaws, and seems to support his needs well.
Searching felt faster than any bugzilla I've ever used. It has this annoying feature of searching in multiple places by default -- e.g. bugs, news items, changelogs -- but this isn't a big problem (and these sections would likely end up empty for us anyway).
Redmine supports custom fields. There is good documentation on how to set them up:
http://www.redmine.org/wiki/redmine/RedmineCustomFields
and that documentation is pretty short. It looks very simple, accessible via the web interface (as gforge's was, which I found nice).
I haven't set one up myself. Seems to require Ruby on Rails, Apache, and a DB. I guess that's a "LAMR" setup =)
Email: Redmine supports managing / commenting / interacting with bugs via email (including email notification when a bug changes). I might stop listing this... seems pretty standard now.
Privacy: not as fine-grained as reported with Bugzilla. Redmine does allow setting privileges best on user classes, so we could for example make it so that users which aren't a 'member' of the VisIt project could, say, comment on bugs but not submit new ones. There's no support for the reverse hash though: we can't say bugID 42 is only viewable by users X, Y, and Z, for example.
Subcriptions: One can 'subscribe' to a particular search as an Atom feed, or integrate bug planning with iCal.
Reports: The 'Summary' listings were useful. It was mostly a page listing some sort of 'category' to a number with the amount of bugs that the 'category' affected. This was easy / not overwhelming to read, and one could quickly click a link to get more detailed information. No auto-generated graphs or anything though.
Time: One can attach estimates of how long a bug will take to fix, deadlines, etc.
I think Redmine could work for us. It seems like it supports everything we'd need, and there's a lot of support behind it so we could probably figure out how to configure it to our likings.
I liked Redmine better than Bugzilla. It seemed a bit faster, and both simpler and more complex. That is, each individual page had a lot of good information on it and less seemingly useless junk. Yet it was also a bit more 'gforge-y' in that it could track commits, had a component for submitting news entries, etc.