Class for catching error and logging
You could use Callable instead of RunnableAbleToThrowCheckedException. Its call method could throw Exeptions and has a return value too. You can use  Void as a type parameter and return null if you don't need these features:

public class MyCallable implements Callable&lt;Void&gt; {

    @Override
    public Void call() throws Exception {
        ...
        return null;
    }
}

TryHelper seems like an Executor to me. I'd name it ErrorHandlerExecutor and rename its tryDoIt methods to submit (similar to ExecutorService). I guess most developers would find it familiar.
errorLoggerExecutor seems a fair name for an instance which logs the errors of the submitted tasks.
I would also rename Handler to ErrorHandler to be more descriptive.