You could split Request() into two separate functions: You call sendRequest(), then go do other stuff, then, some later, you call getResponse(). If you get RESPONSE_NOT_READY, then you know you have to call it later:

#define REQ_WAIT_DELAY     250  // Wait for 250 ms
#define RESPONSE_NOT_READY ((uint16_t) -1)

uint32_t time_request_sent;

void COZIR::sendRequest(char* s)
{
  Command(s);
  time_request_sent = millis();
}

uint16_t COZIR::getResponse(char* s)
{
  if (millis() - time_request_sent < REQ_WAIT_DELAY)
      return RESPONSE_NOT_READY;
  buffer[0] = '\0';  // empty buffer
  int idx = 0;
  // From here, same as original COZIR::Request()
}

You could split Request() into two separate functions: You call sendRequest(), then go do other stuff, then, some time later, you call getResponse(). If you get RESPONSE_NOT_READY, then you know you have to call it later:

#define REQ_WAIT_DELAY     250  // Wait for 250 ms
#define RESPONSE_NOT_READY ((uint16_t) -1)

uint32_t time_request_sent;

void COZIR::sendRequest(char* s)
{
  Command(s);
  time_request_sent = millis();
}

uint16_t COZIR::getResponse()
{
  if (millis() - time_request_sent < REQ_WAIT_DELAY)
      return RESPONSE_NOT_READY;
  buffer[0] = '\0';  // empty buffer
  int idx = 0;
  // From here, same as original COZIR::Request()
}

Update: Here is an example of how this could be used:

COZIR cozir(some_serial_connection);

void loop()
{
    static bool waiting_for_a_cozir_response = false;

    if (time_to_get_another_cozir_reading()) {
        if (waiting_for_a_cozir_response) {
            report_something_is_wrong("This COZIR is too slow!");
        }
        else {
            cozir.sendRequest("Hi Cozir! Pliz send me another reading.");
            waiting_for_a_cozir_response = true;
            /* We won't get the response on this iteration. *
             * We will ask next time.                       */
        }
    }
    else if (waiting_for_a_cozir_response) {
        uint16_t response = cozir.getResponse();
        if (response == RESPONSE_NOT_READY) {
            /* Nothing to do now, really. We will just try again *
             * at the next loop() iteration.                     */
        }
        else {
            waiting_for_a_cozir_response = false;
            do_something_with_the_cozir_response(response);
        }
    }
    do_whatever_else_has_to_be_done_in_the_loop();
}