How does the act of observation influence what is observed?

It depends on how the observation is being conducted. If the observer stays hidden, or the object to be observed is simply unaware, then the influence on the observed will probably be non-existent.  Afterall, one cannot react to something they don't realize is happening.

     If something realizes it's being observed, the change is often instantaneous,especially in unexpected cases.  For example, with animals, if someone makes an unexpected noise too close to them, they immeadiately fly off.  With people, they might immeadiately stop what they're doing, and also possibly run away, especially if they don't know the person.  In Sleeping Beauty, specifically in teh segment where Aurora sings "Once Upon a Dream," she both immeadiately stops what she's doing, and tries to run away upon realizing she's not alone.  (Going on a unnecessary tangent, she's terrible at running away, she just hesitates a few steps further, and it's far too easy for Prince Phillip to catch up.)

     When people expect to be observed, they change, usually to be what they think the observer expects. It can actually be a problem in Psychology experiments, as (at least now), people have to be told they're in an experiment, and people often are influenced by what they hear.  Smith, Bell, and Fusco tested two groups of men in 1986, and told one group that seeing the color pink increases their strength, and another group was told it increased their strength.  Those who were told it would increased their strength showed they had a 10% stronger grip on a dumbell then those told they would be weaker.  Even if they aren't told, the people being observed can often guess, as the observer can send cues that tell the participant what is expected of them, and what the observer hopes to see.  This is defined as demand characteristics, according to Martin Orne.

       Knowing once something knows it's being observed, everything the observer sees needs to be questioned.  The act of observation causes the observed to put up walls, whether physical walls, such as fleeing to put physical space between them, or Aurora's attempt to use a tree trunk as a wall between her and Prince Phillip (seriously, if she was going to use a tree, she should have found a bigger tree further from him...). With animals, the observer now has to wonder, how fast does it normally go along? Where does it usually spend most of its time?  With Prince Phillip, he now has to wonder, does she usually sing to animals?  Is Aurora usually so coy?  Is her main mode of transportation dancing?

         Or, especially with people, the observed puts up mental walls, using societies rules to put out what's expected of them.  Sometimes that influence is a good one.  People who are aware of cameras that are watching and recording their every move in a store are less likely to steal, because they know they'll probably be recognized, caught, and brought to justice.  But it can also be a bad thing, limiting people's true selves because they're afraid of what people will think if they see.