It has all of my pages - 12,845 and updates dynamically as real estate
listings are added and removed from the system. We have a feed into
the site that adds new listings as they become available. As soon as
listings are removed from our IDX, they no longer show up on the
sitemap.

Google is reporting the error below of too many redirects. It looks
like these are due to the delay between Google reading the sitemap and
then actually doing the indexing. On any given day 100 or so listings
could be added or removed.

Goggle webmaster tools has a warning:
"When we tested a sample of the URLs from your Sitemap, we found that
some URLs were not accessible to Googlebot because they contained too
many redirects. Please change the URLs in your Sitemap that redirect
and replace them with the destination URL (the redirect target). All
valid URLs will still be submitted."

What is the best way to deal with this issue? Currently, when listings
are removed from the site,  we do a 302 redirect to my main search
page. The propblem is that Google seems to have a delay between
reading my sitemap and indexing which causes the "too many redirects"
warning.

At any given instant, the copy of the sitemap held by Google should
_exactly_ reflect the _physical_ pages on the site.

I wonder whether you might be better off serving a 404 server response
code for removed pages with a custom error page apologising for the
entry not being there and offering search facilities?  It's less
confusing for the user than going straight to a search page without an
explanation and technically better than a 302, which implies the page
may return in the future.

"News" sitemaps are a lot closer to what's going on with your site
than standard sitemaps are.