"Dry contact" is what a button does—connect two leads together when it's being pressed, otherwise not. Older garage doors simply have a pair of wires for this that gets run to where you mount the button on the wall. You can just splice into that and have the microcontroller connect them when it wants to open/close.

I thought all garage doors had this, but from ratgdo's website I learned that the newer Security+ 2.0 ones don't. Possibly as part of the same money grab to prevent local/third-party; paulgerhardt's comment nicely explains the motivation for that. [1]

[1] https://news.ycombinator.com/item?id=38191712