No, it would not be practical to do that.
It is theoretically possible to convert exhaust into fuel back again. I know rockets use different fuel/oxidizer compounds, like for example hydrazine and its derivatives oxidized by dinitrogen tetroxide, but let's consider a simpler example since the conclusion also holds if applied to all other fuel and oxidizer duos. If we used liquid hydrogen as fuel and liquid oxygen as oxidizer, the product of the combustion is going to be water vapor. It is totally possible to convert the water back into hydrogen and oxygen.
However, the problem with such approach is more fundamental: conversion of combustion products back into substrates (fuel and oxidizer) will require energy input, and more importantly, considering the second law of thermodynamics, it will always require more energy than was obtained from the combustion in the first place. Net result of energy gained vs spent will necessarily be always negative, no matter what method is used to split water into its constituting elements.
Any supposed solution to this fundamental problem would have to violate the laws of physics. And contrary to claims made in some other answers, it is not necessarily hard nor complicated to turn exhaust back into fuel and oxidizer. All water electrolysis requires is two electrodes and a source of electricity with voltage above around 1.23 V; optional, but highly recommended would also be a little amount of strong and electrolytically inert ionic compound like sulfuric acid or sodium hydroxide to increase the rate of reaction. The process itself is really simple, does not require any "cutting-edge" (or whatever buzzword is trendy nowadays) technology and potentially anybody could conduct such process at home right now. And given sufficiently long time, even the products of wood combustion (assuming complete combustion to carbon dioxide, water, and mineral ash) will in some sense partly turn back into wood and oxygen, as growing trees absorb those substances back from air and the ground while undergoing photosynthesis. Such claims could suggest that it could be possible, but difficult and requiring the invention of some innovative technology, while it actually is possible, just would net energy loss and thus not make sense.
It also does not matter what method of reacting hydrogen with oxygen is used: combustion, inverse electrolysis in so called "fuel cell", etc. will always yield less energy than will be required to turn the resulting water back into hydrogen and oxygen.
A famous series of old pictures from 2010 shows a different, but similar in principle, impossible concept (click on pictures to see full-size versions):