LED Blink in assembly with timer1 output compare interrupt not working
I am not set up to test your code right now, perhaps later.  But:
    ; Set the prescalar to 256 in the Timer Control 1B register
    ldi     r16, 0b00000111 
    sts     TCCR1B, r16

When set to ones, the low order three bits of TCCR1B, select an external clock source at the T1 pin, for clocking on a rising edge.  Probably what's happening is that when you finger is near enough the T1 pin you are capacitively coupling mains frequency to it causing it to tick.
/256 prescaler would have 100 in the lower three bits.
If I manage to get a proper environment up to test your code, I'll look into it a bit more thoroughly.