Yes, it's possible and it's been done, in the form of Mixed Oxides (the mix being plutonium and uranium).
Until it went through prolonged shutdowns due to huge technical, safety and economic setbacks (it may have some use up to 2018, if these issues can be resolved), Thorp (THermal Oxide Reprocessing Plant) has been one of several plants that take spent nuclear fuel and process it for use in reactors designed or modified to take it. So, there's an existence proof that Mixed Oxide (MOX) plants can work. Japan was a significant consumer of MOX fuel.
From a theoretical perspective, they ought to be attractive: once-through nuclear fission is only 5% efficient thermally, and less than 2% electrically: that is, of the energy available in the fuel, only 5% is converted to heat, and less than 40% of that heat is converted to electricity: 95% of the available energy is locked up in the spent fuel which is typically destined to be buried as waste, eventually. In reality, the economics and engineering, though technically very clever and supported by very talented staff, suck.
Do bear in mind that the "molten salt" bit is a red herring in this context. Molten salts can be used as the fuel medium, and as the cooling medium, so we need to be careful not to confuse the cooling mechanism with the fuel cycle. In your link, transatomicpower's vapourware is molten-salt-fuel. Reprocessed spent fuel could be solid, molten salt, or liquid salt: real-world MOX fuels are solid. Molten-salt cooling could be used with all sorts of fuels, but designs keep coming back to water cooling because these reactors are complex enough already without adding crazy demands on the cooling side (q.v the gas-cooled reactors). There are proposed fuel cycles that involve the fuel being in the form of molten salts, but that's pretty much orthogonal to whether the fuel is reprocessed spent fuel or not. Several of these molten-salt variants have been tried in experimental reactors, so not all of them are vapourware, but they have been technical and commercial dead ends (see also fast breeders). Conversely, real-world MOX plants operate in several countries today, so have at least had some technical success: the World Nuclear Association estimates that MOX makes up about 2% of nuclear fuel currently, so globally that's about 6GW of electricity from reprocessed spent fuel and material from retired nuclear weapons.