I have a cutting in clear cup with another clear cup inverted on top for the mini-greenhouse.
The cutting is not showing any roots but has small leaves on tip. The edges  of the leaves are
browning and becoming crisp. Is this caused from too much moisture and should the top cup
be removed?

the leave are drying up. that means that the leaves are not getting any water from the root. is the cutting itself starting to dry up? if the cutting is still good. just leave it like it is. make sure the leaves are drying up and not rotting and falling off. as long as it's drying up and falling off and the cutting itself is not drying up, you still might have a chance.

sometimes the first leaves don't make it... I would just make sure the upside down cup has a few holes for new air and extra heat to get out.   and open it each day to give it fresh air.  also, you want to either cover bottom cup with aluminum foil or put it in another  solid cup(with holes in the bottom) to block light from the soil mix.   what I do, is I put a bit of moss covering the entire cutting to block any light that may hurt the new leaves... it works.  and it is easier to remove the moss once you see new leaves.

The first leaves often do not persist, so don't panic. Fresh air is also important, along with humidity.