Few effects, but some
Unlike past editions, becoming undead is not a massive change for a character / monster, but there will be some minor changes.
Do not sleep … or eat. Or breathe.
The undead do not have to sleep, but they seem to continue to get the benefits of a short or long rest, including the ability to use EH to heal, oddly enough. Crawford confirms this here: https://www.sageadvice.eu/2014/09/19/undead-short-rest/
However, this really is not a big change; This is similar to an Elf!
Also, you can not starve, and you can not suffocate.
Some minor immunities
The undead are immune to exhaustion / fatigue and can not be poisoned. But keep in mind that the general immunities that existed in previous additions, including diseases and charms, They are not in 5e.
Healing is more difficult … but not difficult
Many healing spells specifically say that they do not work on undead (such as healing wounds), but Any spell that does not specifically exclude the undead will work, as regenerate or heroes party. And, if you have access to the Xanthar Guide for everything, flood of negative energy It is a spell that specifically cures the undead.
There are also no restrictions on the list of undead that take potions, so feel free to cure them through the use of potions as well.