UART ESD protection, and is it necessary on inter-board connections?
ESD protection typically applies to integrated circuits which are made using MOS technology. MOS (NMOS, PMOS, CMOS) chips are based on field effect transistors (FETs).

The gate of FET's is a layer of insulating silicon dioxide and acts as a capacitor. Any static charge in a human being can be transferred to the gate on the FET, charge accumulation occurs, and thus the electric field builds in strength across that thin layer which makes up the gate of the FET.  Eventually a breakdown event occurs punching a hole through that insulating gate layer, result in permanent damage to the transistor.

It's an effect which affects FET's and not bipolar junction transistors (PNP, NPN).

So if you've got integrated circuits in the design fabricated from MOS technology, then some kind of protection would be good, although MOS integrated circuits tend to have input protection circuits to protect them, but that's really dealing with how the chips are handled before insertion into a circuit board.

I'm thinking about the source of the ESD, where do you think it can come from?
ESD doesn't normally occur.  Are you thinking about thunderstorms and lightening which can induce high voltages into equipment?  I've seen thunderstorms take out electronic apparatus without a direct strike on the equipment, the flash of lightening inducing a high enough voltage in the electronics so as to destroy components.

Electronics with long cable runs are going to be more susceptible, particularly if exposed, located outside of the equipment. If you have cables like this then you might want to think about opto-isolators. I've even seen these fried by a thunderstorm close by, but at least the main printed circuit board and components were protected and not damaged by the lightening.