The similarity in irony between these stories is that someone tells a deceptive story to someone else that has a great effect on their present lives and on the future course of their lives. Irony in this instance comes in at two levels. The first level is in the deceptive stories, The second level is in the effects or results of the story.
The deception told by Vera in "The Open Window" involves irony because she takes information she gets from Framton then uses it against him to scare him out of his wits. In contrast, the expectation is that she would use the information she gets from him to make him feel welcome and at his ease.
"Then you know practically nothing about my aunt?" pursued the self-possessed young lady.
The deception told by the swindling weavers in "The Emperor's New Clothes" involves irony because they prey upon the common human weaknesses of vanity and gullibility to con the kingdom out of wealth in the Emperor's treasure coffers by weaving invisible (nonexistent) cloth. The contrasting expectation is that they would weave real fabric.
this cloth had a wonderful way of becoming invisible to anyone who was unfit for his office, or who was unusually stupid. (Andersen, "The Emperor's New Clothes")
This similarity brings up the important thematic point of motive: What was the motive of the deceivers? In Andersen's tale, the answer to this is an easy one: they deceived in order to get riches (and make fools of the townspeople). The answer to this from Saki's is harder to find. On the one hand, it may be that Vera's motive was simple, unthinking playfulness. On the other hand, it may be that Vera's motive was malicious with secret intent to torment. Saki leaves the text compellingly ambiguous (with no textual evidence supporting one idea or the other) though critics often agree that Vera is the enfant terrible who intentionally torments Framton out of disdain for weak and foolish adults (like her own relatives: "a pack of pariah dogs").
This leads to consideration of the second level: the effects or results of the ironic deceptions. Neither story states the effects or results, but conclusions can be drawn from actions in the resolution. In Framton's case, he runs off in terror. It can be supposed that the remainder of his "rest cure" is a dismal failure. It can be supposed that, ironically, what his sister dreaded is precisely what happens: Framton sits alone without speaking to a sole while his nervous condition grows worse and worse--unless--he is admitted to a hospital for extensive treatment.
In the Emperor's case, the text reveals that, with a "shudder" of cold, he realizes the innocent child is right and he has nothing on. Since he is Emperor, he cannot be removed from office but it is possible some of his wise advisers will be removed, and it is very possible that the Emperor will not stir out of doors again for a long long while. In that regard, he and Framton can be thought of as suffering a similar effect: solitude.