At least, not until specific things occur as a result of being in the cone.
Targets (of spells) PHB 204
Unless a spell has a perceptible effect, a creature might not know it
was targeted by a spell at all. An effect like crackling lightning is
obvious, but a more subtle effect, such as an attempt to read a
creature’s thoughts, typically goes unnotice, unless a spell says
While the Negative Energy Cone isn’t a spell, it is a magical effect, and so it is reasonable to apply the same logic.
Death Tyrant MM 29
The death tyrant’s central eye emits an invisible, magical 150-foot
cone of negative energy. At the start of each of its turns, the tyrant
decides which way the cone faces and whether the cone is active.
creature in that area can’t regain hit points. Any humanoid that dies
there becomes a zombie under the tyrant’s command. The dead humanoid
retrains its place in the initiative order and animates at the start
of its next turn, provided that its body hasn’t been completely
Nothing in the description of the ability describes any obvious effect, other than when trying to regain hit points or dying, so creatures won’t notice it unless either of those two things occur.
DMs may allow checks to notice where the central eye is looking
Given that the cone is emitted by the central eye, a DM may reasonably allow a character to try to determine where the cone is being directed, if said character is aware that such a cone exists. However, this is totally up to the DM as there is no specific guidance on this.
Official rules concur
The Sage Advice Compendium contains additional detailed guidance on perceiving spell effects, so again, the logic applies to our magical effect here:
Do you always know when you’re under the effect of a spell?
You’re aware that a spell is affecting you if it has a perceptible effect or if its text says you’re aware of it (see PHB , under “Targets”). Most spells are obvious. For example, fireball burns you, cure wounds heals you, and command forces you to suddenly do something you didn’t intend. Certain spells are more subtle, yet you become aware of the spell at a time specified in the spell’s description. Charm person and detect thoughts are examples of such spells.
Some spells are so subtle that you might not know you were ever under their effects. A prime example of that sort of spell is suggestion. Assuming you failed to notice the spellcaster casting the spell, you might simply remember the caster saying, “The treasure you’re looking for isn’t here. Go look for it in the room at the top of the next tower.” You failed your saving throw, and off you went to the other tower, thinking it was your idea to go there. You and your companions might deduce that you were beguiled if evidence of the spell is found. It’s ultimately up to the DM whether you discover the presence of inconspicuous spells. Discovery usually comes through the use of skills like Arcana, Investigation, Insight, and Perception or through spells like detect magic.