I don't think "throws IOException" is necessary here, as none of the methods (useCallable, useSupplier and use) declare to throw an IOException. An IOException is thrown from the lambdas, but that doesn't matter.
It's hard to say because there are two lines that are flagged as not compiling. So what would be needed if it were to compile is subject to debate. Which means in this case, if adding the exception makes it clearer, I'm fine with it. (I agree it isn't actually needed).