Would travelling at relativistic speeds have any impact on human biology? If a person was sitting on a craft that has accelerated to near light speed speed from Earth (e.g. 99.999% of light speed) would there be any impact on his or her human biology due to relativistic effects on mass, time and distance?  
For example, would the heart still be able to pump blood effectively if the relativistic mass of the blood has increased?  
Or would accelerating the blood a bit further towards light speed requires more energy than when stationary on earth?
 A: Moving at a constant velocity, no matter how close to the speed of light, has absolutely no effect on the person moving. In fact, it has no effect on the laws of physics. This is the fundamental tenet of special relativity - you cannot tell absolute motion, only relative motion between different things. The changes you are referring to are what someone moving at a different velocity would see from his or her perspective. If you are moving very fast with respect to me, I might say that your "mass" (in the resistance to further acceleration sense) is large, and that you appear squashed in the direction of your motion, but for you it is just a normal day and I am the one who appears to be having problems.
Now two catches. First, if two people start at rest with respect to one another (e.g. sitting on the Earth) and you want to get one of them moving relativistically with respect to the other, there will necessarily be acceleration involved. Human bodies cannot withstand acceleration much beyond the $9.8~\mathrm{m}/\mathrm{s}^2$ or so we feel from Earth's gravity. In fact, a fundamental tenet of general relativity is that acceleration is indistinguishable from gravity, so a large acceleration for a long time would be like living on a planet with a larger gravitational field at its surface, which would probably not agree with your body too well.
Second, if you are thinking of space travel, keep in mind that space is not empty. If you are traveling at near-light speeds compared to your surroundings (e.g. the Solar System), you are likely to encounter some piece of dust that, from your perspective, is heading toward you at nearly the speed of light. Not a pleasant experience. Furthermore, if you go fast enough relative to the Cosmic Microwave Background radiation that permeates the universe, the stuff in front of you will appear blueshifted to the point of being rather harmful (and nearly impossible to shield) gamma rays.
