I am not familiar with those rigs. Which one can be configured as a repeater? I ask out of curiosity, the couple of rigs I have setup as repeaters in the past could be controlled remotely via the key pad on the mic.
As a disclaimer... Are you in the US? Have you read and understand the FCC rules regaurding repeater operation? Guess so or you would not be worrying about remote control. 
I have not used the LM567 either... Gonna be a lot of help, arn't I 
On to the code...
To start lets assume th LM567 is working as expected, pin 8 goes LOW when 1750hz is detected.
For testing use a momentary push button in place of the LM567. Connect a pull up resistor to GPIO.5, a 10K will be fine.(resistor will go to 5 volt) Connect one side of the push button to ground(zero) and the other to GPIO.5. This will make GPIO.5 idle HIGH, when the button is pushed GPIO.5 is forced LOW. Should simulat the LM567.
Connect an LED to GPIO.1 with a 470 resistor so when GPIO.1 is HIGH the LED lights.
Code:
@ device pic12f509,intrc_osc,wdt_off,mclr_off,protect_off
Define OSCCAL_1K 1
'I will assume the config is correct, I have not checked
LOW GPIO.1
START:
IF GPIO.5 = 0 THEN IN_CK
GOTO START
IN_CK:
PAUSE 30000
IF GPIO.5 = 0 THEN LED
GOTO START
LED:
HIGH GPIO.1
STOP
The above is a one shot. If you need to turn the LED off again look at the TOGGLE command, but get the above working and we will go from there.
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