So, one of my players simply refuses to use any common sense and acts with suicidal indifference to obvious traps.
For example: In front of the PC stands a medium-sized scale statue of a dragon. Deep in its mouth, he sees the glittering of a deep blue stone the size of one's fist. In front of the statue lies a pile of dust and some discarded adventuring gear. His Perception is 21, so he also notices that the floor and wall are razed clean of any dust, growth, vegetation or paint in a straight line from the dragon's mouth, to the far wall.
The player then loudly declares, every single time, "I ignore all of this and reach deep into the dragon's mouth to grab the gem!"
Now, I don't have a problem vaporizing people as the DM. I do have a problem with the Rogue plainly stating that they ignore the warning signs, and then putting themselves in a position where there is no possibility for a Dex save. This player will argue, endlessly, that they get a save against the lightning trap, despite the trap setup imposing a condition in which no save is possible. By the time the Rogue has triggered the trap, it's literally impossible for them to dodge anything due to the position they put themselves in. It would be akin to locking their own character in a chest with a delayed blast fireball.
A good answer will have sourced information with respect to mechanics.
Best Answer
"Removing" a save is not (explicitly) supported by mechanics
Let us first consider what your player is doing. When his character moves into a trapped area willingly, he is betting that his reflexes can save him from whatever clever ruses those trap setters put into place. To see similarities of this style used in media, take a look at this clip of the 2011 Three Musketeers film, starting at 0:40.
That said, the player cannot demand a saving throw either
Under the PHB, players do not normally ask to make saving throws.
Saving Throws,
The operating word here is "normally" -- they willingly put themselves in risk of harm, so they expect a save. However, they do not get to demand one from you if you didn't want one.
However, traps cannot always be avoided
This question looks at a similar issue, but that one looks at it from a metagaming perspective. I will quote the list of things a trap can do, however, other than attempt to harm you.
The top answer on that question is very good, so I advise you to give it a read. However, do note that a trap which unleashes a Wall of Force will not offer a save (trap #3), and sometimes when he expects to trigger a trap and you say "You do it, and nothing happens." (trap #4) -- sometimes, that is more frightening.
Traps are expected to have DCs
From the DMG, the section on traps says:
Trap Effects
If your players know this, and their characters have come to expect this, it is only fair to give them a save.
The players might get a kick out of it
Note that Traps Suck, at least according to a certain angry GM, and your players may actually not like traps. Being able to "outsmart" these things with sheer skill/talent/brute force, that they can essentially bypass the puzzle you gave them, is something they might enjoy.
Players love bypassing entire sections of the plot by being outside the box. I know I do. And dealing with traps straight up can seem so "out there" that it might be considered outside the box as well. After all, no normal person in their right mind would trigger that trap willingly.
That said, removing the save entirely will also spoil a bit of their fun, as that was something they were supposed to be good at.
The final levers of control
As the DM, if you ask for a saving throw, you can rule a save to be made:
Should you feel the situation warrants one or the other, or even both, then feel free to apply them. However, you can also of course take the saving throw away as it is within your rights.