"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.
A saving throw—also called a save—represents an attempt to resist a spell, a trap, a poison, a disease, or a similar threat. You don’t normally decide to make a saving throw; you are forced to make one because your character or monster is at risk of harm.
A saving throw can be modified by a situational bonus or penalty and can be affected by advantage and disadvantage, as determined by the DM.
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.
- more damage
- different damage
- traps that actually trap 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:
The attack bonus of a trap, the save DC to resist its effects, and the damage it deals can vary depending on the trap's severity. Use the Trap Save DCs and Attack Bonuses table and the Damage Severity by Level table for suggestions based on three levels of trap severity.
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:
- At a penalty (+/- 2/5 usually)
- With advantage/disadvantage
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.