This is happening because a refresh includes everything - including the postback...  I would consider this a bug of the DotNetNuke feedback module and it should probably be reported.  What should happen is after the postback the server sents a redirect to itself - "resetting" the form so to speak.

A better solution would be to have a weather module that handled its refreshes inside if an iframe or using callbacks (what microsoft came up with years ago but what everyone now calls ajax).  This would remove the requirement for the entire page to refresh regularly, because only the small iframe or xml post/response would be being sent back and forth.  Finding a more powerful weather module that would do this may be your best option.  Alternatively, the DotNetNuke iframes module may have its own refresh options, but i'm not sure.

You could also just disable the refresh for the feedback form specifically, users probably won't stay there long and notice that the weather is not being updated.