The answer to this is very dependent on details. There's many ways to approach this and they lead to very different results.
The first question is whether you ever have infinite energy density. This causes all sorts of equations to break down for the same reasons as a naked singularity breaks down all the laws of physics. Since energy and mass are entwined in relativity, this would also imply infinite mass, which causes even more problems. More than likely the answer to this is "the physics of this universe would be completely and utterly unrelated to anything in this universe." This sort of thing only shows up in the big bang models, and we readily recognize that they're just models to describe what might have happened. If singularities like that were discovered in our universe, it'd be a big deal.
Another option would be to have infinite energy but finite energy density. If you have infinite space, you can pull that off. This can be done in a few ways. One approach is to have the entire universe be "warm," so there is some base energy available at all points permitting change to occur over time. This energy, however, may not be available for doing thermodynamic work. If that energy is in the form of waste heat, then you can't use it for work.
Another approach would be to have the infinite energy in a form which is captureable for work. To do this, we would need to toss away the laws of thermodynamics. The laws of thermodynamics that we are so familiar with only apply to systems that are at thermal equilibrium, and you would be able to ensure the universe never reaches equilibrium. The field of non-equilibrium thermodynamics is a very open field at this point. We're constantly being surprised by results found in that field, so I won't speculate as to what might happen.
You can have the energy density decrease as you get further away from some magical "origin" point. In this case, you could build a universe where you could simply exist forever, because you'd always be able to find an energy gradient, which is the key requirement for doing mechanical work. This would be a departure from our physics because in this world, "we cannot define the position of God's Throne in space." There is no point which is any more central than any other, or any more "fixed" in space.
A final intriguing option would be to change the boundary conditions. We typically assume that energy approaches 0 as we get far away. However, you could have an oscillating boundary condition that has energy going on forever in a repetitive pattern.
This one is rather interesting because we've got examples of it in the world of simulated automata. One interesting class is the 1 dimensional elementary cellular automata. These are simulated automata consisting of a line of cells which are either alive or dead, and change states each timestep based on their state and the state of their two neighbors. They are typically known by their "rule" number, which is a number describing the entire evolution function of the automata. One of these, Rule 110 is particularly beautiful and interesting:
In these pictures, the top row is the "starting state," and each subsequent row is a new generation created by applying the rules of Rule 110. All you have to do for any cell is look at the cell's left neighbor, itself, and its right neighbor, and replace the state of the center cell using the following rule.
current pattern 111 110 101 100 011 010 001 000
new state for center cell 0 1 1 0 1 1 1 0
What's interesting about this is the pattern of small triangles of size 3 that build the "backdrop" for rule 110. This actually appears as long as you have at least 1 "live" cell in your initial state. Accordingly, it is common to explore the movement of the more beautiful triangles on an infinite repeating array of these size 3 triangles.
This would correspond to an oscillating boundary condition. Anything "interesting" in this world can assume that there will be a continuous influx of energy from this osculation which can be harnessed.
As it turns out, Rule 110 is considered to be very special because it is Turing complete! That's right. Anything you can do with your computer can actually be simulated using these little triangles. They make streams of them collide in very particular ways to execute operations using something called a cyclic tagging system.
Incidentally, this is very similar to a rule we see in nature, Rule 30. Rule 30 is:
current pattern 111 110 101 100 011 010 001 000
new state for center cell 0 0 0 1 1 1 1 0
We see this in cone snails, used as camouflage.
So, in short, infinite energy changes everything, but there are ways it can be kept tractable, such as having oscillating boundary conditions. Some of these cases even have really interesting beahviors!