Equivalent function in Firefox developer tools or Firebug to Chrome Resources tab?
Unfortunately, it is not currently featured in the Firefox Developer Tools, but will start with Firefox 34, which will be released in November. See the Storage Inspector section in the following article: https://hacks.mozilla.org/2014/09/webide-storage-inspector-jquery-events-iframe-switcher-more-firefox-developer-tools-episode-34/
As far as Firebug is concerned, some features are already implemented, such as cookie validation, but some others may require additional extensions. See Firestorage Plus for example ! for local and session storage.
Otherwise, you need to use other extensions (not specific to Firebug at all) or methods to check things like application cache or IndexedDB content. It all depends on what functionality of the Chrome resource inspector you are really looking for.
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