Drip can provide some similar information for IE, and Leak Monitor finds some Firefox-specific leak types, but it would be good for browser/add-on creators to provide even more debugging information to authors.
WebKit and Firefox/Mozilla both have publicly visible bug tracking systems. As they work on their respective new versions, regressions will occur that will break web sites, possibly including yours. Often, this is a genuine bug in the browser code, and hopefully it'll be fixed before the final release. However, it can sometimes be due to incorrect assumptions made by your code which no longer hold true when browser code is tightened up. Both WebKit's and Mozilla's Bugzilla instances support generating RSS feeds from search results. By subscribing to feeds for bugs that mention your product name, you can stay on top of such bug reports. For example, here's a shared page (with feed) for the bugs that mention Reader on either site.