Well, it looks like a trend is developing. After some quick research, I found that the HTC Inspire has the Froyo version of Android (2.2). Android 2.3.4 is Gingerbread and the system that immediately followed Froyo. HTC apparently shipped (maybe still ships?) it's own web browser. As far as I can tell, roughly the same browser is in both of your phones. According to a post in this forum:
http://androidforums.com/htc-droid-incredible/218839-best-browser-froyo.html
The refresh problem happens on a lot of websites. So, that's probably a bug in the browser. The posters in that forum recommend installing a browser called "dolphin hd".
I am far from an expert on Android (my total time of using one is about 5 minutes), but I think both of those phones can be upgraded to use the Ice Cream Sandwich with is a much better version of Android. Then, you can use Google Chrome for Mobile which is a very good browser. You might look into doing that.
I don't think there is anything I can do on my side. I know what is going on -- there is a page that is included in other pages that refreshes every 60 seconds to keep your sessions alive. This refresh should be behind the scenes and unnoticeable to users. That's the way it works for almost everyone.