There are several metaphysical approaches to use quantum physics to try to "prove" the existence of God within the current framework of physical knowledge and, of them, this is one which seems among the most intriguing and most difficult to shake, because it's got a lot of compelling components to it. Basically, this takes some valid insights into how the Copenhagen interpretation works, some knowledge of the Participatory Anthropic Principle (PAP), and finds a way to insert God into the universe as a necessary component to the universe.
The Copenhagen interpretation of quantum physics suggests that as a system unfolds, its physical state is defined by its quantum wavefunction. This quantum wavefunction describes the probabilities of all possible configurations of the system. At the point when a measurement is made, the wavefunction at that point collapses into a single state (a process called decoherence of the wavefunction). This is best exemplified in the thought experiment and paradox of Schoredinger's Cat, which is both alive and dead at the same time until an observation is made.
Now, there's one way to easily rid ourselves of the problem: The Copenhagen interpretation of quantum physics could be wrong about the need for a conscious act of observation. In fact, most physicists consider this element to be unnecessary and they think that the collapse really just comes from interactions within the system itself. There are some problems with this approach, though, and so we can't completely role out a potential role for the observer. (Check out the book Quantum Enigma to find out more on this subject.)
Even if we allow that the Copenhagen interpretation of quantum physics is completely correct, there are two significant reasons that I can think of why this argument doesn't work.
Reason One: Human Observers Are Sufficient
The argument being exploited in this method of proving God is that there needs to be an observer to cause a collapse. However, it makes the error of assuming that the collapse has to take prior to the creation of that observer. In fact, the Copenhagen interpretation contains no such requirement.
Instead, what would happen according to quantum physics is that the universe could exist as a superposition of states, unfolding simultaneously in every possible permutation, until such a time when an observer springs up in one such possible universe. At the point the observer potentially exists, there is therefore an act of observation, and the universe collapses into that state. This is essentially the argument of the Participatory Anthropic Principle, created by John Wheeler. In this scenario, there is no need for a God, because the observer (presumably humans, though it's possible some other observers beat us to the punch) is itself the creator of the universe. As described by Wheeler in a 2006 radio interview:
We are participators in bringing into being not only the near and here but the far away and long ago. We are in this sense, participators in bringing about something of the universe in the distant past and if we have one explanation for what's happening in the distant past why should we need more?
Reason Two: An All-Seeing God Doesn't Count as an Observer
The second flaw in this line of reasoning is that it is usually tied in with the idea off an omniscient deity that is simultaneously aware of everything happening in the universe. God is very rarely depicted as having blind spots. In fact, if the deity's observational acumen is fundamentally required for the creation of the universe, as the argument suggests, presumably he/she/it doesn't let much slip by.
And that poses a bit of a problem, because the only reason we know about the observer effect is because sometimes no observation is being made. This is clearly evident in the quantum double slit experiment. When a human makes an observation at the appropriate time, there is one result. When a human does not, there is a different result.
However, if an omniscient God were observing things, then there would never be a "no observer" result to this experiment. The events would always unfold as if there were an observer. But instead we always get the results as we expect, so it seems that in this case the human observer is the only one that matters.
While this certainly poses problems for an omniscient God, it doesn't entirely let a non-omniscient deity off the hook, either. Even if God looked at the slit every, say, 5% of the time, in between various other deity-related multitasking duties, scientific results would show that 5% of the time, we get an "observer" result when we should get a "no observer" result. But this doesn't happen, so if there is a God, then he/she/it apparently chooses consistently not to ever look at particles going through these slits.
As such, this refutes any notion of a God who is aware of everything ... or even most things ... within the universe. If God exists and does count as an "observer" in the quantum physics sense, then it would need to be a God who regularly does not make any observations, or else the results of quantum physics (the very ones trying to be used to support God's existence) fail to make any sense.