I think by 'GADTs' you mean an AST (could be mistaken). In that case, it would not be a bytecode interpreter, it would be a tree walker. In most languages, an AST is translated down to bytecode, and passed to a VM execution engine. How this engine deals with closures is highly compliant on the architecture of the AST.
First, keep this in mind: A closure is just a closure at the higher-level. Job of the AST is to translate it down to ta more abstract form, thus, a closure would most probably be inlined. Otherwise, it will be a global subroutine defined somewhere on the stack. It never makes sense not to inline closures, unless that closure is a 'hotzone', e.g. frequently used, in whcih case you should most obviously define as a subroutine, and if possible, JIT.
A VM lke NekoVM has a higher-order, terse IR with which you can handle closures like you do in JS.
Don't conflate higher-order constructs with intermediate representations. If you are interested to see a VM development in progress, star my RuppVM. Still early though. But i tend to work on it actively.
Thanks.