Are the "mold spots" actually producing spores that you can wipe off or blow off of the surface of the wood or are there just green spots in the grain of the wood? The reason I ask is because I ran into an issue when I stripped and sanded four "support" columns that ring my car port. I found out that the "mold" wasn't mold (I couldn't sand them out either) but rather were green spots caused in the wood from using ready strip and a cleaning solution. I didn't find anything particular about ready strip causing that but I did find that some deck/wood cleaners can cause green spots to show through on the wood because some pressure treated wood has natural green color that occurs throughout it - sometimes in spots or blotching. So you may not have mold - you may just be seeing the wood "naked" which includes green from pressure treatment. It will go away (obviously if you paint it) but if you stain it you won't see the green anymore although the stain may look a little darker or off color slightly where there is green. Hope that helped!

Was steel wool involved in your process at any point? It could well be that the black spots are a chemical reaction of iron with the stripper. They are pretty much indelible.
Casey

"They are pretty much indelible."

Oxalic acid can remove them, but it also bleaches the wood.

All the above is possible. I had a problem a few years ago that after much research and experimentation, I concluded that the wood, finished in Czech Republic, had copper in the stain. It reacted with something in my stripping process, perhaps the zinc in my flow-over tank, and put a blue-green cast over everything. It did not occur when I used a different stripper and stripped outside the tank.

Likewise, it did not come out with bleach or sanding.

"Likewise, it did not come out with bleach or sanding."

Any stain that penetrates deeply will be very hard (and possibly impossible) to remove completely.

The typical black marks in oak from tannins reacting with iron (like the steel in nails) usually end up being the length of the nail and very hard to remove.

If it is just a few places drilling out and plugging with side grain plugs makes a decent repair. Matching older wood can be a real chore unless you can steal matching wood in the house to make the plugs.

Dip tank reactions often go very deeply, especially if high pH (caustics) are used in the tank. The high pH enhances penetration.

Steel wool on oak is almost always a bad idea. The smallest piece of steel will easily cause a black spot.
Non-woven pads are a better option.

Bronze wool used to be used, but the availability is poor and the price is high.

Copper wool is a little easier to find ('Chore Boy pads) but watch out for copper plated steel copies. If it attracts a magnet it is not copper wool.

"Will it show thru when I apply the tung oil?"

Unless the tung oil is very dark it is going to show through.

Use sawdust or non-woven abrasives (Scotch brite pads).

Steel wool and water based 'safety' strippers are a bad combination.

The slightest bit of water can cause steel wool to start rusting in minutes, and any pieces that break off and embed into the wood will cause rust and black marks on many woods.

Well, all the above info seems good. I would caution you about working in directly sunlight however (as you noted about the drying time of your stripper stuff). Plus, when you do decide to put on your finish I would definatley avoid direct sunlight.

I would also recommend that you look into something that I've fallen in love with - it's a product by Wagner and it's called "The Paint Eater". It is an electric power tool that looks like a large flashlight with a round rough pad at the top. That rough pad removes paint, veneer - everything and it does it MUCH EASIER than chemical strippers or even sanding. And the best part is that if you use it correctly it doesn't remove as much wood or cause the uneven "waves" that sanding can cause (especially when you sand one area more than another because it is a tough spot). After you use the Paint Eater you can do some light sanding to ensure you have everything off or use the sanding to get into really tight spaces or contours.

As far as using Tung Oil and will it cover up the stains on the wood. Brickeyee is right in that the stains will show through unless the oil is dark...but I'm not used to just using Tung Oil without a stain first. Are you going to just apply Tung Oil or were you going to stain your piece first then apply Tung Oil?

I wanted to bring out the "blond" of the wood and was not planning on using a stain. I wanted to have a lighter wood finish than I previously had with the maple veneer.