Many circuits nowadays accept only 3.3 V as input voltage, maximum 3.6 V, but they can work often down to 2.7 V.

Since Lithium batteries (LiPo and LiIon) provide 4.2-3.3 V when charged-discharged (not completely, but substantially), a diode could be used to drop the supply by a relatively constant 0.55-0.7 V (depending on current rating of the diode and current circulating) to obtain (for 0.6 V) 3.6-2.7 V.

A capacitor has to be connected after the diode to reduce noise and current peaks.

Provided that the final device accepts 3.6-2.7 V, what are the disadvantages of this solution?

The advantages are minor cost, simpler connections, no quiescent current of a (linear) regulator.

In my specific case I'm thinking about a ESP8266 powered by a 18650 Li-Ion battery: in deep sleep the quiescent current of the regulator (2-3 uA) is significant compared to the chip itself (20-30 uA). A more compact setup is also a good advantage.