Does a charged particle have more mass than an uncharged particle? I was just thinking about Einstein's famous $E=mc^2$, and how it equates energy to mass and mass to energy. I was wondering, if the charge is a measure of energy, then if a particle accepts a charge does it gain mass?
 A: I suppose what you want to ask here is "does charge contribute to the mass of a particle?" and, if one is referring to elementary, fundamental particles like the electron, then I would think it is not possible to answer this scientifically.
The reason is rather simple: in order to be able to determine if charge contributes in the way you're thinking to an electron's mass, we'd need to be able to strip an electron of its charge, so as to have an uncharged version to perform the comparison of masses against. That, in turn, would mean there would have to be a particle identical in every respect (except perhaps mass) to the electron, only missing its charge, i.e. a "neutral electron". 
Otherwise, if we're comparing an electron against some other neutral particle (like neutrinos) it isn't very useful, because such are different particles in many other ways, and hence they do not constitute a suitable experimental control as we have not isolated the independent variable (charge) for which we seek to observe a causal relationship between changes in it and changes in the dependent variable (mass). 
However, as far as we can tell, such "neutral electrons", identical in every way but charge to their charged versions, don't exist. Therefore, we cannot remove the confounding variables, and so there is no scientific experimental study that can be done to assess an answer to this question.
A: If you are asking about color charge, then the answer is no, gluons are known to be massless as per the SM, and they do carry color charge.
If you are asking about EM charge, then the answer is still no, but the explanation is not so simple. According to the SM, at some point, before electroweak symmetry breaking, charged particles were actually massless.

Another take on this issue is that Standard Model particles (and in particular charged ones) were massless before electrosymmetric breaking (at least disregarding other mechanisms of mass generation).

Massless charged particles
But in our world currently, massless charged particles are theoretically not possible, because they would make the matter everything is made up of unstable.

If there were massless charged particles, the electron and positron would become unstable - one problem - and the fine-structure constant would run to α=0 at very long distances - another problem, and it obviously doesn't. So massless charged particles are theoretically impossible in our world - assuming that we empirically know some things such as the fact that there is a limiting Coulomb force at long distances.

So this makes it impossible to compare the same type of particle and see if the charged version has more mass, since we do not know of any massless charged particles in the SM.
Now you could try to find some connection between the net EM charge a particle possesses and try to see if that particle has more mass then the ones with less net EM charge, but again, the down quark has more mass and less net EM charge then the electron, so the answer to your question is no.
A: The $E=mc^2$ formula applies only for high velocities and is not relevant in describing masses. The length of four vectors of particles (and the added fourvectors of systems of particles) is called an invariant mass and does not change under Lorentz transformations.
In the classical picture:
A negatively charged mass has extra electrons to the atomic ones, so yes it will have a larger mass.
A positively charged mass has fewer electrons than needed for the atoms it has, so it will have a smaller mass.
In the quantum mechanical frame the elementary particles have charge as a number that identifies an elementary particle together with the other quantum numbers.  Elementary particles are not composite, so the charge cannot be considered independent of the particle. All macroscopic matter is made up of elementary particles.
So in mainstream physics there is no way charge can be separated from the particle itself to be assigned an extra mass. It might be possible in models assuming that elementary particles are composite but I doubt it..
