Layered perovskite iridates realize a rare class of Mott insulators that are predicted to be strongly spin-orbit coupled analogues of the parent state of cuprate high-temperature superconductors(1,2). Recent discoveries of pseudogap(3-5), magnetic multipolar ordered(6) and possible d-wave superconducting phases(7,8) in doped Sr2IrO4 have reinforced this analogy among the single layer variants. However, unlike the bilayer cuprates(9), no electronic instabilities have been reported in the doped bilayer iridate Sr3Ir2O7. Here we show that Sr3Ir2O7 realizes a weak Mott state with no cuprate analogue(9) by using ultrafast time-resolved optical reflectivity to uncover an intimate connection between its insulating gap and antiferromagnetism. However, we detect a subtle charge density wave-like Fermi surface instability in metallic electron doped Sr3Ir2O7 at temperatures (T-DW) close to 200 K via the coherent oscillations of its collective modes, which is reminiscent of that observed in cuprates(10,11). The absence of any signatures of a new spatial periodicity below T-DW from diffraction', scanning tunnelling(12) and photoemission(13,14) based probes suggests an unconventional and possibly short-ranged nature of this density wave order.