There is no concept of controls in ASP.NET MVC any longer.
You have two options:
When the user clicks a button, you handle this POST request in a controller action, set some sort of flag in your view model to now show a textbox, then return the same view which in its turn will look at the flag and generate a text box if required. This will incur a complete round-trip which will be somewhat similar to the postback in WebForms.
You do it in-place with JavaScript. You intercept a click event on a button and you inject an input/textarea HTML element into your document structure.