There is no such thing as 'infinite mass', which is why it's impossible for an object with nonzero invariant (rest) mass to accelerate to light speed in the first place. You can't pour unending amounts of energy into a system to make it move at c, because unending amounts of energy don't exist.
Also, only a bit unrelated to your question is bringing the quantum fluctuations of the type associated with virtual particles into the argument. Because, even if an infinite amount of energy could manifest due to such fluctuations (it can't, as far as I know), the energy would have to be 'repaid' to the vacuum within the limit of uncertainty of the measurement, which is a very short interval. In other words, it's not "usable" energy.