PIC10F206 MPLAB X IDE v5.40 - Light a LED
From the 10F206 data sheet:

The  Output  Driver  Control  register  is  loaded  with  the contents
of  the  W  register  by  executing  the  TRIS  f instruction.  A  ‘1’
from  a  TRIS  register  bit  puts  the corresponding   output    driver
in a High-Impedance mode.

Your program attempts to make GP1 an input, not an output.
In this particular processor, the register TRISGPIO has no address, so you cannot use a movwf f type instruction to set GPIO direction. You must use the specific instruction tris 6...it moves the contents of W register into TRISGPIO.
Or tris GPIO might also write to the TRISGPIO register too, since a header file would likely #define GPIO 6.
About the LED...its anode would face the PIC's Vdd pin, cathode faces GP1. Add a series resistor - something in the 100 to 1000 ohm ballpark.
I used Microchip's MPLAB IDE 8.92 to assemble a corrected version of your program. It includes a nice single-step simulator that lets you see internal bits
   #include p10f206.inc                          ;registers, config bits defined here
        __config _CP_OFF &amp; _MCLRE_OFF &amp; _WDTE_OFF ;MPASM compatible
    ;---------------------------------    
        ORG 0x0000
        
   INIT:
      MOVLW b'11111101' ;make GP1 an output pin
      TRIS GPIO         ;move content of working reg to TRISGPIO
      MOVLW b'00001101' ;set GP1 output low so that the LED with pullup res. lights up
      MOVWF GPIO        ;move working reg to GPIO
     
      movlw 2           ;select GP1 pin for toggling
  
  LOOP:
      xorwf GPIO   ;toggle GP1 pin
      GOTO LOOP ;loop forever every 3us
    ;----------------------------------
        END ;needed to end the program


Note that other PIC processors do GPIO direction control in different ways. In others, the direction register does have an address, where you can movwf f .
This one is a bit of an oddity.