!!! 25+ ESTIMATED SUCCESSES!!! (Either reported or from thanks on this post)
For TL;DR - start at [[ WILL THIS HELP YOU? ]]
This ABSOLUTELY fixed my AT&T Note 3's GPS. However:
:::: DISCLAIMER :::: I had an unusual chain of events leading up to this fix. I hope it fixes your Note 3's GPS, but there's really no way to tell - unless you decide to try it. It may not help you at all. So, what have you got to lose? Well, take a look at [[ WILL THIS HELP YOU? ]] and [[ RISKS ]].
[[ INTRO ]]
If you're here, most likely your Note 3's GPS doesn't work, and you're desperate like I was for a fix. Here's the chain of events that led to my fix:
1. Got AT&T Note 3 - **out of the box** it had the 'blurry' camera issue, but outstanding GPS (consistent 3 second satlock).
2. Sent Note 3 in for Samsung Warranty Repair to fix the 'blurry' camera.
3. Samsung fixed the camera.
4. Disappointingly, Samsung broke the GPS (by bending the 'leaf' pin circled RED in my pic below, I assume inadvertently during reassembly).
5. Got Note 3 back - camera fixed, GPS broken by Samsung.
6. After observing an iFixit teardown, I followed this chain of logic: A - Since Samsung replaced my camera, and B - the camera is separate from the main board that the GPS is on, then C - Samsung must have only replaced the camera, and left the GPS alone. D - Therefore, Samsung did not replace my exceptional GPS, that, sadly, is now broken. E - Something else must have broken the GPS. F - I read a thread suggesting that tightening the 12 screws on the back fixes the GPS, then G - I decided it's time to check on the leaf pins after looking at the iFixit teardown. Below is what I found.
[[ WILL THIS HELP YOU? ]]
Hopefully. I have no idea. Maybe. Decide if the [[ RISKS ]] are worth it to find out if one of your 'leaf' pins are bent or damaged (mine was bent by Samsung Warranty Repair, as explained above).
[ 1 ] Yes, if one of the 'leaf pins' is bent, dislodged, or otherwise not making contact.
[ 2 ] Yes, if one of the contacts that the leaf pins touch is oxidized. (Remove oxidation by scraping it off.)
--- Thanks to evilpotatoman for reporting this fix! ---
[ 3 ] Yes, if one of the 'leaf pins' is squished and is not pressing hard enough against the contact. (Bend "leaf pin" outward to make tighter contact.)
--- As suggested by evilpotatoman ---
Also, perhaps adding a small piece of stock paper under the offending leaf pin will make it keep contact better. ( !!!! Be careful - don't add too much paper, which could stress the pin or contact !!!! )
--- As suggested by superdookie67 ---
 Yes, if a screw is loose. This may happen if you remove the back cover frequently, which loosens the screw that keeps a solid connection between the leaf pin and contact near the back cover fingernail hole.
--- As suggested by yedidi2006 ---
Let's be honest: since Samsung Warranty Repair dislodged my Note 3's leaf pin, isn't it plausible that a maybe even a few Note 3's shipped with bent leaf pins?
I'M WILLING TO BET YES. So please reply and let everyone know if this fixes it.
[[ RISKS ]] (Honestly, this is a pretty basic procedure. But here goes)
1. Potentially voiding warranty, though I have no idea how it can be proven (put some new Loctite on the screws afterward if you're paranoid.)
2. (Only a risk for those who are completely careless) Potential damage to internal equipment.
3. Losing 30 minutes of your life. Personally, I love tearing stuff apart whenever I have an excuse, so for me this wasn't a risk.
[[ INSTRUCTIONS ]] (See attached pics for details)
1. See your favorite teardown site for even more details. Remove the back cover, stylus, SIM, and microSD.
2. Remove the 12 Phillips screws on the back. They are sealed with Loctite, so don't damage your phone with too much force.
3. !!!Carefully!!! remove the inner plastic shell (clipped in).
3a. !!!Only lift up!!!, perpendicularly from the screen to avoid damaging the leaf pins.
3b. !!!Take your time!!! Gently open around the edges until all edges and clips are loose.
4. Observe the leaf pin locations shown below.
5. Samsung bent my leaf pin circled in RED below. This one is clearly related to the GPS, because bending it back restored my GPS to perfect functionality.
6. !!!!Extra Carefully!!!! bend the leaf pin back into position, if you find a dislocated leaf pin.
7. Replace the inner plastic shell, making sure to not damage the leaf pins.
8. Replace the 12 screws (with new Loctite, at your discretion).
9. Obviously, replace the SIM, microSD, stylus, and back cover.
10. Fire it up.
11. MOST IMPORTANTLY, reply back if this helps - maybe this is a fix for many of the Note 3's that shipped with terrible GPS reception.