Introduction and simplifications
Most previous answers predict the end of the world. And they may probably be right. But let just assume it would work. For the sake of simplification, let us assume that for the time of the experiment, the country is isolated. So no taking over from a foreign power, or economic retaliation. The international diplomacy and economy is frozen to what it was before the voting mode was changed.
We also need to remember, that if a certain relation with power is often there, most people without prior background (like father/brother having been elected president already) start in politics with the force of conviction and the will to actually do something beneficial for their country and fellows. They do not start in politics because they want to be rich. Wall street's much more efficient for that.
We can further simplify, and that is not too far fetched, to assume that there are mostly two parties, and thus two candidates for the POTU job. Interestingly, said parties could elect one person who, for a given time would take the job if their party got the majority. They can't organise elections everytime it happens. For even more simplification, we'd take a direct election system like the French have: you directly vote for the president. Or for the party. Not for local representant, which complicates the whole process.
So similar to currency market trade, I would assume that a substantial amount of votes would change everyday in both directions. Much like a lot of euros are exchanged against dollars everyday as well as the other way around. Again, with the same image in head, the dollar-euro parity changes if, on the whole, more exchange is made in one direction or in another. If we elimiate other currencies, 1 euro being at 1.25 dollars mean that the Americans buy more European goods than the European, Americans'. Very similarly, a party who is in favour would preceptively get more vote than the other. The evolution everyday, would make the ratio of votes of the one against the other evolves everyday.
Changing job needs time. Of course, with such a system, people have to be ready to leave on a moment notice, but nevertheless, I would limit the effect by having a count everyday. It would be very costly to have people switch for 5 minutes.
Political leaders, who would want to stay in office, would limit themselves to short-term, popular policies. Anything long-term and it would be undone by the next guy, or it could change twice before any result come out. Unpopular measures, and you're sure to be out by the next day.
The parties would organise at all time a lot of propa... erm, information: to try and convince voters that the other guys are a bunch of useless idiots, and maybe to get some popularity for some measures. While the former is done by parties during political campaigns (extending currently to almost 2 years before the election with pre-primaries, primaries, and election campaign) and would be permanent, the latter is usually done by government. I can give at least two examples: the Death Penalty in France was still favoured some years prior to its abandon. But the government organised some information to convince the people of the problem. And then it got removed. Something similar was done for nuclear power in Germany in the 2000s. It is not necessarily bad. It is just that people opinion have a certain inertia, and rulers usually have access to some information before the rest of the population. And it does not always work. Anyway, this is something that takes time, and it would then be organised by political parties instead of ruling governments.
Other effects and conclusion
Another, maybe unexpected to some, effect, would be that due to the time required to get to know all the subjects, assessors and high civil servants would be more powerful. Indeed the leaders may change from one day to the next. But someone has to keep track of what's going on. And they are the persons who would stay in place.
On a longer term, I can see two possible outcome: either the change is very frequent, and people will get bored, knowing that at the end the ones I mentioned in the last paragraph are the ones in power, and you can't decide anything about them. Or changes are less frequent, and people are checking the evolution, but nevertheless, the society would be pretty rigid, as the effort to make any structural reform would be very costly (in term of money -progaganda- and time).