Are there be any Halachik issues with regard to using someone else's logo, twitter avatar or the like as part of an app?
If it would be okay, would that only be if I don't make any money from the app?
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The opinion of Rabbi Moshe Feinstein and Rabbi Shlomo Zalman Auerbach is that infringing on intellectual property is a violation of hasagat gevul -- competing with someone in an unfair way -- and thus prohibited.
I am not an intellectual property lawyer, but I'm strongly inclined to say that whatever you have in mind, if it's legal in your country (assuming your country has solid intellectual-property protections), halacha would allow it as well. If it's illegal in your country, it would also be prohibited by halacha, both out of dina demalchuta dina (respecting the law of the land) and hasagat gevul.
So please contact appropriate sources as to the legality of your intellectual-property question.