A search for the production of heavy partners of the top quark with charge $5/3$ is performed in events with a pair of same-sign leptons. The data sample corresponds to an integrated luminosity of $19.5\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ and was collected at $\sqrt{s}=8\text{ }\text{ }\mathrm{TeV}$ by the CMS experiment. No significant excess is observed in the data above the expected background, and the existence of top-quark partners with masses below 800 GeV is excluded at a 95% confidence level, assuming they decay exclusively to $tW$. This is the first limit on these particles from the LHC, and it is significantly more restrictive than previous limits.