Why does the tempo tend to get faster? Suppose you're learning to sing or play an instrument, and you'd start playing some piece of music. If you are not experienced enough, the tempo would be unstable, usually getting faster until the end.

This phenomenon is usual, as in this question on Music SE: https://music.stackexchange.com/questions/32405/why-i-tend-to-play-faster-tempo-without-realizing-it

Many correlates it with adrenaline and throbbing heartbeat, but is there an experiment-backed evidence to explain?

(Originally posted in Skeptics SE, but I think this should be the better place.)
Tempo should be an objective parameter in music. The same piece is heard differently at higher tempo than in lower tempo. This is sensible if you accept an Aristotelian perspective of music, whereby music is an attempt to represent something real (could be a person's emotions or natural sounds). The key in your question seems to be "not experienced enough". Such a musician can lose focus of the real, intended character behind the piece (if we follow the same Aristotelian perspective). He is likely to focus on short term aspects such as: Grasping the current melodic pattern or focus on his own ability to perform the piece. Focusing on your grasp of melody will make you tend to play faster since your fascination will come from the melody itself, and not from what it represents, or could represent. If you are interested in pushing your ability to perform, like many beginners are (for good reason), you might still tend towards a faster pace since it is simply more challenging neurologically, and might boost the so called "muscle memory". 
In any case, altering the tempo is altering the piece, and no skillful music performer is supposed to do that against the composer's wish!
My own experience is both from playing and composing, even though I have only been doing the latter for quite some time now.