If all that was done was to use the API without including any of Telegram's code (not even header files) in the application, traditionally the answer would have been that nothing needs to be published since nothing subject to the GPL would be included in the application. Depending on rulings in the Oracle v. Google case, that answer may change.
If actual code from Telegram is included in the application, then yes the entire application would have to be distributed under the terms of the GPL unless the copied code can be isolated to an external program that's merely used by the application.
There's a slightly grey area when it comes to only having included header files (function and data structure definitions without any implementation code present) in the application. The grey depends on whether those header files are considered simply facts about the API (not eligible for copyright) or a creative work in and of themselves (eligible for copyright). This is usually more relevant when using software libraries, and since that usually means the application would be linked against the library and thus would include it the answer usually ends up being yes the application needs published under the GPL. For a network API such as Telegram's, a simple description of the data format was usually considered more factual than creative and depending on the language used for the client the issue might not even come up if the headers couldn't be used directly. Again, depending on the final rulings in the Oracle v. Google case this area may be subject to change.