I wouldn't know what would accelerate the planet this much without destroying it as a side-effect. Not to speak of any life on it. But I can say that if it manages to, I expect some serious side-effects.
For the record: high-relativistic speeds are pretty crazy. If I neglected or misinterpreted one of their effects, well, this is just a best shot without spending too much time, since nobody else seems to give this a serious try.
Illuminated habitat at solar spectrum
When moving in the vastness of space, the CMB (cosmic microwave background) actually makes up the majority of incoming radiation. This site has a nice plot and some explanation on that. So, at least in terms of energy, starlight won't be the major factor there.
Now, as to a habitat, I think this will go bad if I understand black-body radiation correctly. I will ignore incoming particles for now and tackle a basic problem of the question: both the CMB and our sun are following Planck's black-body radiation. This means that, if we want to get the spectrum that the sun emits, we have to use a black-body source corresponding to the same temperature, or else the shorter wavelengths' intensities will drop significantly, giving us less visible light, and more infrared and beyond. Red- or blue-shifting black body radiation doesn't change its properties, only its temperature. So, to get the kind of light we have, a large portion of the forward-facing sky (before aberration) would radiate with the intensity of the sun!
The sun has a solid angle of just $7 \cdot 10^{-5}$ as seen from Earth. If the equivalent of something on the scale of a quarter of the sky is a black body at solar temperature, this giant sun would burn the planet. This rules out having light as on Earth, we'll have to go to longer wavelengths.
To make the power of our incoming CMB equal to that of the sun on Earth, we can equate the incoming intensity times solid angle of both of them. (Again, I am assuming aberration just compresses the light into one direction, and work with the undistorted sky for now.) This way, we can calculate the black-body temperature the CMB would have to replace the sun in heat output. Using the Stefan-Boltzmann law for the scaling of power with black-body temperature, we get $T^4 \pi = T_\odot^4 A_\odot$, where $T$ is the desired temperature, $T_\odot$ the sun's temperature, $A_\odot$ solid angle, and the $\pi$ is the equivalent of a quarter of the sky.
(The value for the corresponding effectively illuminated sky is just a rough guess, but should be good enough. See the comment by verlaner. Note that the visible image of the sky may be heavily distorted due to the aberration of light, bundling the incoming light into the forward direction.)
With $T_\odot = 5800\text{K}$ and the solid angle from the previous paragraph, this gives $T = 398\text{K}$.
That's merely about $125^{\circ}\text{C}$. This black-body temperature is far too low to give the desired illumination, but already heats the planet at the full power the sun would yield.
Velocities
I have the feeling that the idea to blue shift the CMB into the sun's spectrum is not healthy in general. Let's say the CMB magically doesn't harm us, but we need it to have the solar spectrum, to see just how much velocity this is.
The CMB has a peak wavelength of about $1\text{mm}$. The sun is at $0.5\text{µm}$. So you want to get a factor of about $2000$ in frequency.
The relativistic Doppler effect gives a frequency ratio of $\sqrt\frac{1 + \beta}{1 - \beta}$, where $\beta = \frac vc$. This yields something like $v \approx 0.9999995c$.
Let's calculate the kinetic energy per mass at this velocity.
$$
\gamma = \sqrt{\frac 1{1 - \beta^2}} = 1000
$$
$$
E_\text{kin} = mc^2 (\gamma - 1) \approx 9.0 \cdot 10^{19} \frac{\text J}{\text{kg}} = 90 \frac{\text{EJ}}{\text{kg}}
$$
So, severe time-dilation aside, obstacles are bad. One gram of infalling mass would have a yield of $90\text{PJ}$, which is roughly 1500 Hiroshima bombs. I guess it's safe to say that dust and meteorites will be a much bigger threat for this planet. Also, if the planet is quickly slowed down due to collisions, nobody would want to live on it. ;)
Assuming it is as large as Earth, the planet is sweeping up volume at $r^2\pi c \approx 3.8 \cdot 10^{22} \frac{m^3}{\text{s}}$. If it passes through a molecular cloud with $10^7 \frac{\text{hydrogen molecules}}{\text{m}^3}$, this yields about $10^{23} \text{W}$. For comparison, our sun inputs about $10^{17}\text{W}$. This doesn't sound healthy.
However, outer space -- outside of galaxies -- has densities below one atom per cubic meter. This strips the six orders of magnitude we need to get it below solar output. I admit I don't have any idea whether this kind of bombardment might deal damage to atmosphere or habitat directly, even if its absolute power isn't high.
Not that any of that is relevant compared to the black-body radiation argument above. All in all, my estimate is that this will not end well, and probably quickly too.
No warranty on correctness. Please comment or edit if you find a mistake.
