I don't like those photo-eyes either... it's not too risky a
proposition to get rid of them. I guess until the neighborhood kid gets
stuck under it. Anyhow, here's my 2cents, FWIW:

I'm familiar with digital electronics, but not garage door openers, so
I am making some educated guesses:

You'll want to limit the current to about 20mA max (typical of digital
circuits)....

So, let's assume the input on the garage controller is a sinking-type
input (ie it looks for positive voltage). The photoeye most likely is a
Normally Closed type (so if you were to cut the wire, it would default
to the emergency state, ie something in the way of the beam)... it most
likely uses a low-voltage source, maybe 24V, maybe 12V, or even 5V.
You'll have to check into this. Using Ohm's law we can figure out the
right resistor based on the current we want (20mA)... V=IR; R=V/I. For
24 V, R=24/.020=1.2KOhm.... 12V, R=.6KOhm... 5V, R=250 Ohm.

So, try tying the input to Voltage with the corresponding resistor in
series.

Good luck.