Is there a way to check whether an app explicitly supports Resume or not?

Most "Lion-Ready" apps will also support the new full-screen mode, so if you see the little full screen button, it is likely the application will also support Resume. However, there is no definitive list that I am aware of yet, nor is there a way (beyond a mention in the app's documentation) of knowing whether it will support Resume.

However, there is now a list of "Enhanced for Lion" apps on the appstore which might be a good place to start.

Activity monitor has a column with yes/no for apps that support sudden termination which should be a good proxy for resume functionality.

The combination of bmike & lemonginger's answers may be better than either one alone.

bmike is referring to a developer API that was added in Mac OS 10.6 (and therefore may not be a perfect proxy for the full resume support).

From Apple's developer documentation:

Mac OS X v10.6 includes a new mechanism that allows the system to log out or shut down more quickly by, whenever possible, killing applications instead of requesting that they quit themselves.

(You may need a developer account to view this document. I'm not sure.)

He suggests it because support for "sudden termination" is a prerequisite for Lion resume support. As bmike suggests this makes it a reasonable proxy for support even if it is not a guarantee of support.

leonginger's is basically suggesting that an application that has been updated in any way for Lion is more likely to include resume support. This is a faulty assumption on its own but becomes more powerful when combined with bmike's suggestion.

Logic: if an application has obvious signs of having been updated for Lion (such as fullscreen support) AND the application has support for sudden termination the chances are quite high that that application supports resume.

The new (in 10.7) NSWindowRestoration protocol is the entry point for restoring windows. Due to Objective-C's dynamic nature, it won't be a perfect test, but there's probably a very good correlation between an application containing a reference to this symbol, and that application supporting Resume.

$ grep NSWindowRestoration "/Applications/QuickTime Player.app/Contents/MacOS/QuickTime Player"
Binary file /Applications/QuickTime Player.app/Contents/MacOS/QuickTime Player matches
$ grep NSWindowRestoration "/Applications/Photo Booth.app/Contents/MacOS/Photo Booth" 
$ 

If this symbol is present, it means the programmer at some point in the program at least mentioned the name of the new API which is only used for Lion Resume. This isn't a guarantee of anything (it could even be a string that says "I don't support NSWindowRestoration yet, sorry"!), but it's good circumstantial evidence.