Chosen Solution

I am not sure if I should clean this before reinstalling. If it is adhesive I don’t want to ruin the home button. I noticed that the gasket seems in good shape and the home button was easy to get removed. Very little resistance without using heat when taking it off carefully. Should I replace the membrane if the adhesive isn’t sticky when reattaching the home button to a new screen?

Definitely clean it! This is human detritus. Other than being gross, it won’t cause any problems per se but I would always clean that gunk away before re-assembling a device.