There's 5 pages of replies, I didn't read them all, if someone else mentioned this then consider this post redundant.
The problem here is that the Higgs Boson giving particles mass is one of those "I don't think that means what you think it means" situations.
First, the Higgs Boson itself gives nothing mass. Instead, what gives SOME particles mass is the interaction with the Higgs Field (called the Higgs Mechanism). Essentially think of it in this simple analogy: As the particle moves through space, it interacts with the Higgs Field which provides something like drag on the particle which slows it down and prevents it from moving at the speed of light. That drag-analogue is what we call mass. The problem is, of all of the particles in your body, of all of the particles in the chair your ass is firmly planted to, of all of the particles that make up most of the universe only the electrons are effected by the Higgs Mechanism. Go back and reread that last sentence, it's important. All of the electrons in your body weight somewhere between 20 and 30 grams depending on how big you are. Yes, only 20-30 grams of your TOTAL MASS (about 0.03% of your total mass) comes from the Higgs Mechanism. Neutralizing the Higgs Mechanism would do nothing to facilitate faster-than-light travel unless all you were moving were electrons.
The other 99.97% of your mass is being provided by protons and neutrons. What you need to understand is that electrons are fundamental particles. They aren't made of anything smaller. Protons and neutrons are NOT fundamental particles. They are instead made up of quarks and quarks are NOT affected by the Higgs Mechanism.
The quarks interact with what's known as the Gluon Field. This interaction causes the same drag-analogue as in the Higgs Field, but it only affects quarks, and in turn, protons and neutrons. But the quarks themselves are only responsible for about 1% of the total mass of a proton so where does the rest of the mass come from? Well, something to know about quarks is that they require a massive amount of energy to bind together, and in fact you can never separate quarks. You'll never be able to have a single quark by itself, but that's another topic all together. Long story short, E=mc^2 here means that all of that energy in the system creates yet more mass.
So, there's no way to ever neutralize the mass of an object because 1) There's no way to make E=/=mc^2 and 2) if you neutralized the effects of the Gluon Field (somehow) you would then cause all of the quarks in the area to cease to exist and without the quarks there's no protons or neutrons and you'd suddenly be left with a cloud of electrons.
I'm a sucker for quantum physics and read, watch, and absorb all I can on the subject.
Sources for reference:
https://www.youtube.com/watch?v=J3xLuZNKhlY
https://www.youtube.com/watch?v=Ztc6QPNUqls