Apart from Philipp’s practical answer, we can also approach this problem theoretically. Both the $\ce{Li+}$ ion and the $\ce{Li}$ atom have electrons in lower energy levels that can be excited to higher levels given adequate heat and that can re-relaxe under photon emission to their original energy level.
In the case of a $\ce{Li}$ atom, we see an emission at $670~\mathrm{nm}$ corresponding to an energy of $1.85~\mathrm{eV}$. This relaxation corresponds to a $\ce{2s<-2p}$ transition; i.e. two quantum states with the same principal quantum number $n$. We need to add at least this transition’s energy if we want to excite an electron, since it is the lowest possible excitation energy.
In the case of a $\ce{Li+}$ ion, we can at first approximation assume the energy levels of a helium atom. Thus, the lowest excitation should be $\ce{1s->2p}$ ($\ce{1s->2s}$ is forbidden), including a change in the principal quantum number $n$. This corresponds to an energy of $\approx 20~\mathrm{eV}$. That corresponds to a wavelength of $62~\mathrm{nm}$ — well in the ultraviolet range, already approaching x-ray wavelengths. That would mean two things:
Undergraduate labs would likely disallow this experiment due to the very harmful wavelength (the longest-wavelength line of the Lyman series of hydrogen is $121~\mathrm{nm}$, just for comparison).
You would need much more energy to excite the electrons than a simple lab flame would make available. Remember that a bunsen burner flame has a temperature of $\approx 1300~\mathrm{K}$; according to this image on Wikipedia that is not really hot enough to truely emit ultraviolet photons in any reasonable concentration let alone come close to $62~\mathrm{nm}$
By comparing this to the experimental result (‘we can colour the relatively cold flame with lithium salts’), we see that the assumption ‘$\ce{Li+}$ ions are responsible for the flame colouring’ must be rejected.
Note: All calculations and predictions presented herein are based on some kind of layman’s simplification and may be physically inaccurate.