you can communicate telepathically with your familiar while under the effects of Feign Death
Feign death has the following effects:
- You touch a willing creature and put it into a cataleptic state
- For the spell’s duration, or until you use an action to touch the target and dismiss the spell, the target appears dead to all outward inspection and to spells used to determine the target’s status
- The target is blinded and incapacitated, and its speed drops to 0
The incapacitated condition states:
An incapacitated creature can't take actions or reactions.
The relevant section of Find familiar allows you to:
While your familiar is within 100 feet of you, you can communicate with it telepathically. Additionally, as an action, you can see through your familiar''s eyes and hear what it hears until the start of your next turn, gaining the benefits of any special senses that the familiar has. During this time, you are deaf and blind with regard to your own senses.
The paragraph, read as a whole, is clear. Communicating telepathically with your familiar does not require an action. It would, however (since it is communication) require you to be conscious.
Now, to determine if you are conscious, we need to figure out what a
is, since the spell specifically puts you into that state (the blinded and incapacitated conditions are in addition to this state).
Using a variety of dictionaries and relevant scientific definitions, a cataleptic state is defined as:
a physical condition usually associated with catatonic schizophrenia, characterized by suspension of sensation, muscular rigidity, fixity of posture, and often by loss of contact with environment
catalepsy is defined as a state of marked loss of voluntary motion in which the limbs remain in whatever position they are placed.
(Science Direct: https://www.sciencedirect.com/topics/neuroscience/catalepsy)
a trancelike state marked by loss of voluntary motion in which the limbs remain in whatever position they are placed
a medical condition in which a person's body becomes stiff and stops moving, as if dead
None of these definitions require a lack of consciousness on the affected party (though both the dictionary.com and Merriam-Webster definitions potentially suggest mental faculties can be affected).
As a result, nothing in the rules precludes you from telepathic communication. Therefore you can do this.