The function of little is the same in all of these sentences:
There is little evidence that whole fruit is good for you.
There is little evidence that whole fruit is bad for you.
There is little support for my proposal.
In short, negative and positive outcomes have nothing to do with the use of little.
However, there is a difference between little and a little.
There is little evidence of X.
This means that there is almost no evidence. It's saying there is essentially an absence of evidence. Whatever evidence does exist is not enough to support X.
It's not X that's negative, but the assertion of a lack of evidence that's negative.
There is a little evidence of X.
This means that there is actually some evidence. That there is some evidence means that it's possible to assert a claim of X. There may not be enough evidence to support it conclusively, but there's enough to not rule it out entirely.
It's not X that's positive, but the assertion of the existence of evidence that's positive.
Normally, we would use a little if we want to stress something positively, and just little if we want to stress something negatively. Typically, it's a good X we want to put a positive spin on and a bad X that we want to put a negative spin on:
Thankfully, there is a little evidence [positive] that hugs [good] are effective.
Thankfully, there is little evidence [negative] that spankings [bad] are effective.
But it doesn't need to be that way:
Alas, there is a little evidence [positive] that eating live worms [bad] is healthy.
Alas, there is little evidence [negative] that eating doughnuts [good] is healthy.