It will be fixed in build 1251, see:
http://www.ventismedia.com/mantis/view.php?id=5703

Give us the ability to clear the cache for specific feeds, or just have it do it automatically if there's an error.

We can hardly do it automatically, because we cannot know whether the link is unavailable only temporarily or forever.

Give us an "ignore" feature on the error box.

The error box should be here only once with indication to disappear in 10 seconds, isn't it?
And at the end of the whole process there is one error log.

Give us a "remove this item from cache" item in the error box.

This couldn't work fine in case there is another problem (like temporary server unavailability)
Note that you can remove the links manually from the feed by using Del key without a need to resubscribe.

Nevertheless you seem to be right that error reproting is annoying when the update podcasts action is performed automatically.
I would also suggest to keep the error reprorting disabled until user updates podcast manually.
It worked this way up to build 1245 and then we changed it
See discussion here:
http://www.ventismedia.com/mantis/view.php?id=5585
especially notes 0017789 and 0017792

Would changing it back meet your needs?

A few updates ago, a feature was added to show an error message when you have a problem downloading a podcast. This is good in theory, but it's been very annoying having to deal with it.

A few of my podcasts generate "incomplete download" messages, but the files that are trying to download don't actually exist in the feed, just in MM's list of files. I'm guessing that this is caused by MM caching the file list from a feed, and then later that feed is updated with a different file name/location, and the incorrect file location is never removed. There are two entries for certain podcast episodes: one for the correct file, and one for the incorrect one. When I unsubscribe/resubscribe to a feed, the old incorrect files are removed.

However, this will continue being a problem in the future, and I don't want to have to unsubscribe/resubscribe each time a podcaster gets his file location wrong. (Especially since that causes MM to download each episode in the feed.)

Some suggestions to fix this:

Give us the ability to clear the cache for specific feeds, or just have it do it automatically if there's an error.
Give us an "ignore" feature on the error box.
Give us a "remove this item from cache" item in the error box.

Seeing the error box every time I update my podcasts (and every hour automatically) gets pretty annoying.

Thanks for your consideration. I've been a long-time MM user, and it's still the best media program I've ever seen. Keep up the good work!