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Here is the original code:

int ledpin = 13;     //activates arduino light when there's a detection
int ledpin2 = 11;    //activates external LED or buzzer when there's a detection
int outputpin = 3;   //pin the relay control is attached to
int sensorpin = 12;  //pin the detector is attached to
int warnpin = 4;     //pin used to activate warning LED or buzzer
int counter = 0;     //countdown since last detection
int counter2 = 0;    //countdown since last non-detection

//change the next 2 settings to adjust the timer intervals
int countmax = 10000;  //maximum level of countdown timer. 1000 = 1 second
int countwarn = 2000;  //when to turn on warning pin. 1000 = 1 second

void setup() {
  pinMode(ledpin, OUTPUT);
  pinMode(ledpin2, OUTPUT);
  pinMode(outputpin, OUTPUT);
  pinMode(warnpin, OUTPUT);
  pinMode(sensorpin, INPUT);
  counter = countmax;
  counter2 = countmax;
  //any line with "serial" is for debugging and should be commented out if not needed because it will slow the clock down
  //Serial.begin(9600);
  //Serial.print("count: ");
  //Serial.println(counter);
  //Serial.print("count2: ");
  //Serial.println(counter2);
}

void loop() {
  if (digitalRead(sensorpin))  //what to do when there is a non-detection
  {
    digitalWrite(ledpin, LOW);
    digitalWrite(ledpin2, LOW);
    counter = counter - 10;
    counter2 = countmax;
  } else  //what to do when there is a detection
  {
    digitalWrite(ledpin, HIGH);
    digitalWrite(ledpin2, HIGH);
    counter = countmax;
    counter2 = counter2 - 10;
  }

  //serial output lines for debugging
  //Serial.print("count: ");
  //Serial.println(counter);
  //Serial.print("count2: ");
  //Serial.println(counter2);
  //Serial.println("-----------");

  //sound a warning if either counter is above 0 but under the "countwarn" level
  if (counter > 0 && counter < countwarn || counter2 > 0 && counter2 < countwarn) {
    digitalWrite(warnpin, HIGH);
  } else {
    digitalWrite(warnpin, LOW);
  }

  //turn off the output if either counter drops below 0
  //these settings are for an "active low" relay.  Switch "HIGH" and LOW for an active high output
  if (counter < 0) {
    digitalWrite(outputpin, HIGH);
    counter = 0;
  } else if (counter2 < 0) {
    digitalWrite(outputpin, HIGH);
    counter2 = 0;
  } else {
    digitalWrite(outputpin, LOW);
  }

  delay(10);  // wait time (miliseconds) before going again
}

If you simply do not connect an external LED or buzzer, the code will do exactly what you want - just turn on/off depending on the detector.