I am looking for a generic answer. Lets say I have built mobile app running on Android and iOS platform and REST API's for these apps lie under https based web application.
Will it cause any speed or performance issue when these mobile apps try to fetch data from this web app? If I host this web app under http, will these apps perform better?
I am not concerned about server performance, just concerned about how mobile apps will behave in such case?
Will it cause any speed or performance issue when these mobile apps try to fetch data from this web app?
Not significantly. The decryption overhead will be swamped by other effects, such as network I/O, parsing the payload of the response, etc.
You can see this by visiting Stack Overflow using https, then with http (ideally using a fresh "incognito" browser window each time, to eliminate local caching effects). You should not notice much of a difference. That's because parsing the HTML/JS/CSS and rendering the page, plus the network I/O to get all that stuff in the first place, is what takes up the bulk of the time.
If I host this web app under http, will these apps perform better?
Not that the user should notice. Hence, either:
Run your own tests to confirm the behavior for your specific use case, or
Default to using encryption (https), if you do not wish to run those tests
