Putting together an AR isn't the most complicated thing to do. That said, it does take at least a bit of ability and knowledge of the basics to get it right, especially on the upper side. I've been playing with ARs and M4/M16's long enough to have seen the most common points of failure.
Lower:
Springs installed incorrectly causing light primer strikes or locking pins to work their way out.
Leaving the disconnector out completely causing the trigger to not reset after a shot. (don't laugh, I've seen it)
Detents installed backwards or left out because they got lost during installation.
Trigger guard mount ears broken off due to not being supported while driving in the roll pin.
Scuffs, scratches and gouges from not protecting the lower while driving roll pins in.
Upper:
Improperly torqued barrel nut.
Using loctite instead of grease on the barrel nut making it damn near impossible to ever get off again.
While over-torquing the barrel nut can make it hard to take off again later, severely over-torquing it can actually bend or shear off the alignment pin on the barrel extension. If the alignment pin bends, the whole barrel can be canted in the upper causing mis-alinment of the bolt lugs. This will damage or destroy both the extension and the bolt. It can also cause feeding issues with the feed ramps being out of alignment. The upper can also be bent or cracked from over-torqing.
Barrel headspace being improperly set from the factory and not checked by a novice/lazy builder.
Using an M16 barrel extension on a lower with M4 feed ramps . (not as common anymore since the vast majority of barrel extensions and lowers have M4 feed ramps now)

Bolt Carrier Groups:
This is probably where you will see the biggest variations in quality.
Some companies stake the gas key screws by denting the material on the top of the gas key. Not a good way to stake those bolts IMO.
Good: (staked from the sides)

Not so good:

Horrible:

Some bolts are also made from better materials than others and any bolt you expect to be reliable should be MPI tested. Chrome lined carriers are the industry standard these days too.
Bolts with fancy chrome, Nickel Boron, etc coatings are nice but not necessary. BCGs don't really wear out quick enough to make it worth worrying about. Coatings can make them easier to clean, however.
Some good info on BCGs here:
http://www.weaponevolution.com/forum/sh ... ier-GroupsBarrel/gas blocks:
Gas block not properly aligned, pinned or secured. This can cause all kinds of issues. It usually won't damage anything, but it can cause short-stroking, front sight cant or otherwise unreliable function. Some gas blocks with set screws require dimpling the barrel so the screws have something to "lock into" otherwise they can move around or loosen up as they heat up from firing.
Clamp-on blocks can be hard to keep aligned and clocked since they tend to move a bit when the clamping screws are tightened. (just like scope mounts)
Some companies might drill the gas ports larger or smaller. Too small will under-gas the rifle. Too large a port can potentially over-gas the rifle but that can be compensated for using weighted buffers, adjustable gas blocks and heavier bolt carriers.
With frankenrifles, you have to trust the builder matched all his/her components to work together properly. Not hard to do if you stick to quality brands and standard spec parts, but you're always risking having to replace parts to make such a rifle run smoothly.
I'll add more as I think of it.