Does turning propane or butane gas tank upside down help to use the remaining gas?
In principle it won't make any difference, but in practice it sometimes can.

Since what you want to emerge from the tank is gas, in principle nothing you do will make more gas come out. The liquid inside will vaporise until there is no gas left and the pressure won't change while there is still liquid in the vessel (both gases liquefy easily under pressure and the pressure will be relatively constant while there is still liquid.) All this is true at equilibrium and constant temperature.

However, in practice the assumption that the inside of the vessel is in equilibrium and constant temperature is not always true. If you use a lot of gas, the vessel will cool as the liquid evaporates (and you sometimes see a ring of frost on the outside of the vessel corresponding to the level of the liquid). If this happens then shaking the vessel may have the effect of warming the liquid and increasing the pressure of the gas. So shaking it might help a little.