One of the principles in ckassical physics is that inertial reference frames are equivalent - in Newton's physics, this means that absolute velocity cannot be determined. You have probably already heard this. (Acceleration can result in a pressure.)
There are crystals which vibrate differently when they are put under pressure - the pressure squashes the crystal changing the spacing between the atoms.
If a constant velocity produced pressure, then it would be possible to tell what velocity you are going by observing the way the crystal changes it's "tone". It's a simple experiment with cheap components... observing that tone-change would overturn hundreds of years of established physics.
But that's all Galileo and Newton stuff - special relativity is odd isn't it?
In special relativity, the "time dilation" is what one observer sees of
another observers clock.
If Alice and Bob are moving as relative speed v, the A will see B's clock time dilated and B will see A's clock time dilated. Neither notice anything special about their own clocks.
i.e Bob could look at Alice and, not knowing about SR, could deduce that there is some influence on Alices clock making it slow down. He could assume that Alice's clock is under some sort of pressure and, after a lot of experiments, work out how the pressure varies with speed. This would be a valid way to investigate the phenomenon.
This does not prove the clock is under pressure from the speed - it just investigates the relationship between the observed dilation and the speed. The trick in science is to try to come up with an experiment to disprove the idea that speed produces a pressure.
The first sign comes when Alice is stationary and Bob is moving - Bob still sees the same time dilation happening to Alice's clock! In order to feel the right amount of pressure to show the right amount of dilation, Alices clock needs to know how fast Bob is going. How can this be?
So Bob asks Alice about it - it's her clock so she should know what's "really" going on. But Alice denies that anything funny happened to her clock. Instead, she says, it is Bob's clock that has been running slow all this time.
We check all these things with atomic clocks, and verify the resulting models with other clocks - like the half-life of muons. Have you seen the Mt Wellington muon experiment?
http://www.scivee.tv/node/2415