Is disabling built-in apps fine? I am using Moto E(1st gen) with Android 4.4.4 KitKat. It is unrooted.

I wish to disable some of the built-in apps, which I do not use, e.g. Motorola Assist, Motorola Migrate, Google Play Books etc., to free up internal storage and increase the smartphone speed. 

Whenever I try to disable a built-in app, the following warning pops up: "If you disable a built-in app, other apps may misbehave. Your data will also be deleted."

Is it safe to ignore this warning and proceed with disabling it? Or does it really cause the problems it mentions? Also, can disabling these apps have the reverse effect of slowing the phone down, further?

I looked up questions related to this in Android Enthusiasts Stack Exchange, and found these : How can I prevent unwanted pre-installed apps from automatically running on Android?, and How can I stop applications and services from running?, but they do not exactly address this issue.
Most apps on android are safe to disable, however some can have some pretty bad side effects.  This however depends on what your needs are. You can disable the camera for example but it will also disable the gallery (at least as of kitkat and I believe Lollipop is the same way).

Android does have some core applications that are essential to the function of the actual system itself. For example disabling Package Installer will disable the ability to install any application and will likely cause your phone to go into a boot loop/soft brick. 

More to the question, if you are unrooted you should be relatively safe since I can't recall any application without root privileges that can access and 'take down' any vital apps on the system.  That being said going mucking around with the core applications that make your phone/tablet function shouldn't be done lightly.

Source: I build custom roms, and I am a former PAC ROM maintainer.