Can crown cement get lodged in gum tissue?

It's not common for the cement to get embedded into the gums to the point you need surgery to clean it out. Usually it's right around the margin of the crown. It is pretty common to have cement left behind, but not to the level you're describing.

Hi
Yes it can and it should be removed because it can cause gum disease if it stays there. 

Hi There,It is not common to have cement in-bedded in gums.  It should be gotten out of there.  That is a foreign object, like a splinter and can cause pain and damage to the surrounding tissues.