The presence of planetary material in white dwarf atmospheres, thought to be accreted from a dusty debris disc produced via the tidal disruption of a planetesimal, is common. Approximately 5 per cent of these discs host a co-orbital gaseous component detectable via emission from atomic transitions - usually the 8600 Å Ca II triplet. These emission profiles can be highly variable in both morphology and strength. Furthermore, the morphological variations in a few systems have been shown to be periodic, likely produced by an apsidally precessing asymmetric disc. Of the known gaseous debris discs, that around HE 1349-2305 has the most rapidly evolving emission-line morphology, and we present updated spectroscopy of the Ca II triplet of this system. The additional observations show that the emission-line morphologies vary periodically and consistently, and we constrain the period to two aliases of 459 ± 3 and 502 ± 3 d. We produce images of the Ca II triplet emission from the disc in velocity space using Doppler tomography - only the second such imaging of a white dwarf debris disc. We suggest that the asymmetric nature of these velocity images is generated by gas moving on eccentric orbits with radially dependent excitation conditions via photoionization from the white dwarf. We also obtained short-cadence (≃4 min) spectroscopy to search for variability on the time-scale of the disc's orbital period (≃hours) due to the presence of a planetesimal, and rule out variability at a level of ≃1.4 per cent.