Does $E$ really equal $mc^2$? If mass is sped up to the speed of light, does the mass become energy? I'm currently in a debate with a co-worker.
Does $E$ really equal $mc^2$?
If mass is sped up to the speed of light, does the mass become energy?
 A: No it doesn't. $E = m c^2$ holds only for massive objects at rest. When they move this relation no longer holds. The more generic energy-mass-momentum relationship does hold in these situations:
$$E^2 = m^2 c^4 + p^2 c^2$$
In this equation, $m$ is the invariant mass, $E$ the total energy, and $p$ the momentum. When the object with invariant mass $m$ speeds up, $E$ and $p$ both increase, while $m$ stays the same as what it is at rest. Hence its name 'rest mass'.
A: This is too naive, as far as physics goes, question and the title asks a different question than the body.  The short answer is yes for "Does $E$ really equal $mc^2$", and no for "If mass is sped up to the speed of light, does the mass become energy".
This is due to the fact that there are two uses of the word "mass" in physics, depending on context.
Read on:
Rest mass is a characteristic measure of matter. When we talk of elementary particles mass is the rest mass of each, $m_0$ which does not change no matter how fast it is going. 
$E=mc^2$ either means $E=m_0 c^2$ for an object at rest, or $E=m_{rel} c^2$ when the object is moving.
The equation $E = m_{rel} c^2$ holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same. 
It is the relativistic mass which grows with the velocity, 
$$m_{rel}=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$
and is due to the kinetic energy given to the particle, the rest mass does not change. From the formula it is evident that a massive particle can only incrementally approach the velocity of light, never equal it; even approaching it, enormous amounts of kinetic energy would be needed. Further reading in the wiki article.
Edit after comments in recent answer by @Ron Maimon .
Studies in nuclear and particle physics as well as in astrophysics have established that even though our lives are lived in non relativistic environments, relativity rules from the microcosm up. This view becomes organized by the use of four-vectors, which distinguishes  two masses:
a) into an invariant part,$m_{0}$ that characterizes a body, in the same way that a length characterizes a ruler in three dimensions, 
and b) a relativistic mass $m_{rel}$ that includes the energy that a body has due to motion.
At rest the two masses are equal, and in our everyday world where velocities are much smaller than $c$ indistinguishable experimentally.
In this language all mass is energy,  from $E = m_{rel} c^2$, 
one sees that  $E/c^2 = m_{rel} $
for all velocities, since $c$  is constant. So the answer  is that a material body is energy from 0 speed on, and this  does not change for very high speeds except as increase in energy. Only the kinetic part of this energy is available in the non relativistic everyday world.
Thus the real physics answer is neither yes nor no.
A: No.
Mass can be converted into energy(it happens in e.g. the sun and in a nuclear power plant), and energy into mass(e.g. the big bang).
An other form of energy is motion energy(called kinetic energy) it increases as the object is accelerated toward the speed of light(the maximum speed).
A: Yes, $E=mc^2$ always, for everything, because what we call "mass" colloquially is just the energy contained in something. The quantity physicists call "mass" nowadays is the mass as measured when travelling along with something, and this is the rest-mass. But the rest-mass is not what weighs on a scale, nor is it what you feel as resistance to pushing, nor is it additive to give the mass of a composite system, nor is it the source of gravity. The energy (divided by $c^2$) is all these things. When the energy is converted in units using $c$ to become a mass, it is called the "relativistic mass", which, when you choose units so that $c=1$, is just a synonym for energy.
Lev Okun has argued that the concept of "relativistic mass" is outdated, and pedagogically useless. He is completely wrong on this. Once a student understands that what you call "mass" day-to-day is really the energy (divided by $c^2$), that frees up the useful short word "mass" to mean something else, namely "rest mass", but not until you understand the equivalence of energy and colloquial mass.
A: Logically, the answer is "Yes", but only because of the way logical implication is defined; see here.
If mass is sped up to the speed of light, then sure, mass does become energy (or a bowl of petunias, or the square root of existentialism).
The point is that the premise is not possible; speeding mass up to the speed of light would require infinite energy input.  Not just energy equivalent to the mass, via E = m c^2, but quite literally infinite.
The statement "If I'm 20 feet tall, then my name is Fred" is true, because I'm not 20 feet tall (the fact that my name isn't Fred is irrelevant).
But the answer to a better-phrased version of your question would probably be "No".  No matter how fast you accelerate a mass, the original rest mass is still there.
