That depends on the app you're disabling. If you disable Google Play Services, it absolutely can mess up the phone and cause things to not work. If you disable Facebook there shouldn't be any issue.
That depends on the app. If it's a system app, you'll get that notice simply because you are changing the stock programming in a way that the manufacturer didn't intend. Some system apps being disabled can be good (i.e. Verizon will install one that adds extra bookmarks to Chrome that can't be deleted unless you disable it). Some apps being disabled can prevent the phone from carrying out necessary functions.
Is there a specific reason or app you are wanting to disable? Just going through and randomly disabling anything and everything can have negative consequences. It would also help if you told us the make and model of the phone (including if it's a carrier branded version).
Re: Does disabling a app on a phone take away memory space on it
Well, those are really 2 different questions. By "memory space" do you mean RAM for an active app or storage space. Also by "take away", do you mean use more or use less?