How to identify the output of dex:get as being JSON or HTML in Common Lisp?

@ignis volens is right, you need to use the Content-Type header. There is some work to do to update the code.

1. Using defparameter, part of the Dexador response is discarded

If the function dex:get is called in the REPL, it will return 4 values:

* (dex:get "http://ip.jsontest.com/")
"{\"ip\": \"xx.xx.xx.xxx\"}                       ;; value 1: IP address
"
200                                               ;; value 2: HTTP code
#<HASH-TABLE :TEST EQUAL :COUNT 6 {100A0EA203}>   ;; value 3: HTTP headers
#<QURI.URI.HTTP:URI-HTTP http://ip.jsontest.com/> ;; value 4: URI

When using defparameter, only the first value will be stored in the variable.

* (defparameter response (dex:get "http://ip.jsontest.com/"))
RESPONSE 
* response
"{\"ip\": \"xx.xx.xx.xx\"} ;; value 1: IP address
"
*

One way to save all the values could be to use the function MULTIPLE-VALUE-LIST

* (defparameter response (multiple-value-list (dex:get "http://ip.jsontest.com/")))
RESPONSE
* response
("{\"ip\": \"xx.xx.xx.xx\"}                           ;; value 1: IP address
"
 200                                                  ;; value 2: HTTP code
 #<HASH-TABLE :TEST EQUAL :COUNT 6 {100A146813}>      ;; value 3: HTTP headers
 #<QURI.URI.HTTP:URI-HTTP http://ip.jsontest.com/>)   ;; value 4: URI

2. Read the Content-Type header from the response

One could write a function to get a field from the HTTP headers, based on the DESTRUCTURING-BIND macro.

(defun get-dex-header (dex-response header)
  (destructuring-bind (raw-response http-code http-headers quri)
      dex-response
    (gethash header http-headers)))

The use of the function is straight-forward:

* (defparameter json-response (multiple-value-list (dex:get "http://ip.jsontest.com/")))
JSON-RESPONSE
* (get-dex-header json-response "content-type")
"application/json"

* (defparameter html-response (multiple-value-list (dex:get "http://bing.com/")))
HTML-RESPONSE
(get-dex-header html-response "content-type")
"text/html; charset=utf-8"

My answer would go also into the direction of @Robert's answer. But I would use multiple-value-bind. How about a function content-type-get which is like a get but in the first position returns the value of the header's content-type.

(defun content-type-get (url) 
  (multiple-value-bind (text http-code http-headers quri x) (dex:get url)
    (values (gethash "content-type" http-headers) 
            text 
            http-code 
            http-header 
            quri 
            x)))

;; this in-addition-content-type-returning get function you can use in return with
;; `multiple-value-bind`. The advantage is that you request the response
;; only once - which ensures speed.

(multiple-value-bind 
  (content-type text http-code http-headers quri x) 
    (content-type-get "http://ip.jsontest.com/")
  (cond ((string= "application/json" content-type) 'do-sth-with-json-text)
        ((string= "text/html; charset=utf-8" content-type) 'do-sth-with-html-text)
        (t 'or-something-else)))

For instance

(defun dex-dispatch (url)
  (multiple-value-bind (content http-result headers uri) (dex:get url)
    ;; cope with errors perhaps here
    (handle-content-type (intern (string-upcase (gethash headers "content-type"))
                                 (find-package "KEYWORD"))
                         content http-result headers uri)))

(defgeneric handle-content-type (content-type content http-result headers uri))

(defmethod handle-content-type :around (content-type content http-result headers uri)
  (declare (ignorable content-type content headers uri))
  (if (= http-result 200)
      (call-next-method)
    ...))

(defmethod handle-content-type ((content-type eql ':application/json)
                               content http-result headers uri)
  ...)

...