Yes. You should tell the players when a creature is dead, unless it is faking it.
Hit Points are an abstraction. They do not measure how wounded the creature is, just how close to being killed it is. A perfectly healthy dragon could die from a single arrow in his chest. But the hit point mechanics simulate the wearying of the combatants. That is why the fighter's second wind is able to recover HP: he is not really wounded, just winded.
This is also the reason the combat capacity of creatures and characters do not go down with HP, as is the case in other systems where being wounded can bring penalties.
A dragon with 1 HP left is as dangerous as a dragon at full HP. It is just easier to drive a killing blow to.
So unless the creature has some way of feigning death, the characters will know it is alive.
A dead monster is easy to spot: it stops moving, attacking, and reacting. Remember that while combat has those static miniatures, in the simulated world the creatures are moving, dodging and feinting all the time. Also breathing and looking around for other hidden threats. All of that stops upon death.
It is recommended in the PHB that most monsters and creatures should die outright when reaching 0 HP (page 198 or SRD 5e link):
Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws.
Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.
The important part is to have consistency. The DM should choose one way or another and keep the same rule throughout the game.
If you require an action to verify if a creature is dead, it should be a very easy to moderate Medicine test, depending on the physiology of the creature (telling an orc is dead is easier than telling a gelationus cube is dead).
And if it is faking, there should be a saving throw (in case of magic or a spell) or a contested roll of deception vs insight.