The First Thing We Do, Let's Kill All The Clerics
With apologies to Dick the Butcher, if you want your characters to feel the heat in combat and stop taking clerical healing for granted... kill the cleric. Or more properly, take out or somehow occupy the cleric early in the combat, so they can't easily dispense healing at the earliest opportunity in a combat.
On the one hand, intelligent adversaries who can recognize clerics and want to actually win the fight would do this as a matter of course. Assassins certainly qualify, here, and ought to be well-briefed on who they're trying to kill.
On the other hand, one expects this to work maybe once or twice before the group starts stocking up on healing potions or otherwise taking protective countermeasures. (And if the opponents are focusing fire on the cleric, that means they're probably not focusing fire on the wizard, the barbarian, or whoever the biggest and most reliable damage-dealer in the party is.)
A Frame Challenge Or Two
But remember, there's a natural narrative tension running through most D&D games: Most D&D games have a lot of combats, and most D&D games aren't grimdark enough to be killing characters on a regular basis. And if you run through enough combats where everyone survives, eventually the player are going to figure out that maybe-- just maybe, whether it's because of the existence of healing potions, or clerics, or because of the way the whole game world is structured-- they're not quite treated the same way as the minor NPCs.
That's why clerics and healing potions exist. They are meant to be used to rescue the characters.
But not all tension in a game comes from the immediate threat of death in any given combat. Another way to look at things (both at the level of an individual combat, or a series of related combats before the group can rest, regroup, and recover their resources) is as an exercise in resource management.
At the level of an individual combat, every time the cleric has to go over and heal someone (and remember, those healing spells have a range of touch, meaning the cleric has to go over there consuming movement and action(s) on a round) means that he or she can't be doing anything else that might be useful, like buffing other characters or casting damage-dealing spells against their opponents.
And at the level of a series of combats, every healing spell cast now is a healing spell that cannot be cast in the next combat, or the next after that... until the players shut down and take a long rest. This is critically important to the way 5e and related systems work. Your PCs can feel like demigods if they blow through scarce resources in the early (probably less challenging) combats of a sequence, only to be left high and dry when fighting the Big Bad at the end.
If that's what's happening, they'll eventually feel the pinch.
If you're following this school of thought, then one of your jobs as GM (which is not necessarily easy) is putting the characters in situations where they can't come at each combat at full strength like a sledge hammer, but rather where they'll have to run through a gauntlet with some real uncertainty on their part of where they should be spending those resources.
A natural result of this-- and I think a good one-- is that not every combat will feel like a mortal threat, but some of them toward the end of those sequences, surely will.