How much tire clearance is needed for your front tires to your fenders at its closest point? I had 205/75R14 tires on the car and went to 205/70R15. At the closest point during the turning process my clearance went from 5/8" to 5/16" with the new tires. Is that to close? How close have you guys ran without an issue before?

That clearance can change depending on your ride height and hitting bumps or low spots when turning, like driveways & such.
5/16" doesn't sound bad depending an how you're checking it.
Closest point should be bottom corner of fender in front of the tire.
That's actually easy to trim back at a 45-degree angle with tin snips and will barely bother the paint if you're careful.
I've done that on a couple of Dusters and early B-bodies.

Car is sitting on the garage floor and I bounced the front end up and down my hand to make sure front end is settled after putting the tires on. I than turned the wheel left and right and checked clearances. The closest point is the bottom corners of the fenders, everywhere else there is plenty of room. Just didn't know if 5/16" is enough under normal driving conditions.

depends on where that 5/16" is. If it's to the sidewall it will rub in corners. If it's to the tread, well tread growth occurs. I have hte same issue with my 65 Cuda and a bit of judicial body work added a bit more (pull lower corner out by hand, trim excess metal , etc.).

Bouncing the car by hand doesn't really duplicate road conditions, unless you're able to fully compress the suspension, but:

Typical interference will be at the edge of the tread/sidewall, not the sidewall itself.
1/2" has always been fine for me, but if I was rallying or road-racing the car, I'd want more.
Another way you can gain more clearance is pushing the fender away using the fender brace; usually one end is slotted.
But like I said, I usually trim that sharp corner back with some snips, touch up the edge, and go.

Here's the closest point. I will try and massage the fender lip some to see how much that gets me. There is no fender brace on the car, doens't look like there ever was one, its a 69 Barracuda, so maybe they didn't put them on them?

If that's at the closest point, with the tires pointed straight, no problem. The tread clears by more when the wheels are turned, right?
Wheel-opening moldings complicate the trimming a bit.
I assumed it was a Duster.
My last A-Barracuda was many years back.
If the fenders have a tab at the bottom with a hole in them, and there's a hole in the bumper brackets, a brace connects there.
Dusters & 68-70 B-bodies have those braces.

Yeah no hole in the fender or bracket so this must be pre fender braces time. The wheel is actually turned slightly in the pic, maybe 1/2 a wheel turn to the right. This the is the closest point at any time. While the wheel is straight or turned more there's plenty of clearance, it's just right at this particular angle that its that close.

i know pics can be deceiving, but it looks ok here. just take it kind of easy at first just in case. if the middle of the tread of that tire [when turned left] has more clearance, you are good to go.

The good news is that as long as the car is moving forward, the tire will push the fender/chrome out of the way, should it make contact.
The bad news is it will be pretty ugly if the wheel shifts forward when you are backing up, it will tend to pull the chrome into the wheel and that will trash the chrome trim. Be very careful backing up.

Personally, I would see if I could get that extra 1/4" of clearance. Move the fender forward, or trim or bend the bottom corner for the extra clearance. Gene