On long enough cosmological time scales, hydrogen and helium nucleii will become scarce in the Universe. It seems to me that any advanced civilisations that might exist in that epoch would have the motivation to try and prevent the stars from using them up, in order to burn the fuel more slowly and extract a greater proportion of the energy as usable work.
One might say that, on time scales measured in trillions of years, the stars are an unsustainable use of the universe's fuel. This question is about whether such civilisations would have the means to do something about it. My questions are:
What would be the most energy-efficient way (using known physics) to blow apart a star or otherwise prevent or greatly slow the rate at which it performs fusion? We're assuming this civilisation has access to vast amounts of energy but doesn't want to waste it unnecessarily, since the aim is to access energy from the hydrogen the star would have burned. In order for this to be worthwhile, the energy gained from doing this would have to be substantially more than the energy the process takes.
What would be the astronomical signature of such an activity? If it was happening in a distant galaxy, would we be able to detect it from Earth?
I'm still interested in this question. There are some great answers below, but currently I don't feel that any of them approach the most efficient way to solve the problem, i.e. a way likely to be attempted by a civilisation with the resources and motivation to do so. Luboš' answer makes it clear that the energy requirements are not too stringent in themselves (you just need to skim off a small fraction of a nearby gas giant and fuse the hydrogen into heavier elements; this question already presupposes the ability to extract power from fusion on a literally astronomical scale) but points out that no "bomb" made of atoms can enter a star and blow it up. However, this does not rule out other methods, such as increasing the star's angular momentum through some means (as in AlanSE's answer) or somehow (perhaps electromagnetically) removing plasma from its surface rather than blowing it up from the inside.
If removing plasma from the surface is possible then ideas based on this have a certain appeal, because the resulting hydrogen could be fused, releasing energy that could be used to skim off more, and so on exponentially. (Kind of like the opposite of that bit in the film 2010 when a self-replicating monolith turns Jupiter into a star.) Some of the resulting energy could be put into increasing the star's angular momentum as well. However, I don't know enough about plasma physics to know whether there's a way to suck the plasma up from the star's surface. (It can be done gravitationally, of course, but then the hydrogen just ends up being depleted by runaway fusion on the surface of another star, so this isn't very useful.)
Given this, my current specific questions are:
Is there a plausible way to remove material from the surface of a star at an appreciable rate, either electromagnetically or through some other means? If so, is there anything other than the (virtually unlimited) availability of energy that limits the rate at which this could be done?
Given a sufficiently huge supply of energy, is there an effective way to rapidly increase a star's angular momentum? It seems (per AlanSE's answer) that redirecting the star's radiation would take too long, but perhaps firing massive objects or particle beams at it would accomplish the same effect in a shorter time.
Aside from these ideas, would adding heavier elements than hydrogen to the star suppress fusion by absorbing neutrons? What quantities would be required in order to affect the star's evolution by this method? (A civilisation that eats stars will generate a huge supply of stable nucleii as a waste product, so maybe it makes sense to simply dump some of them into the next star.)