Is it important to use special fasteners in treated wood for non-structural applications

This should be a good starter project, here are a few tips I've picked up over the years:

• Even pressure treated wood will deteriorate over time if it's exposed or in direct contact with the ground/moisture. Do what you can to protect it from the elements and get it raised off of the soil. Use a moisture barrier between wood and concrete.
• Take care to perfectly level the area you are locating this.
• Make sure your cuts are straight, preferably using a square to mark your lines.
• Measure twice, cut once.
• Look for the direction that the wood bows, mark the bowed up side on the wood, and layout all the wood with the bowed side up. You don't want the floor to be wavy or to have a dip in the center for water/dirt to collect.
• Assemble the sub-assembles first, driving your nails/screws from the side of the joists into the spacers.
• Make sure all connections stay perfectly vertical since they lose their strength when they start to tip over. You can use a clamp to get leverage and untwist a warped joist, but if the warp is too significant, I'd cut it into pieces for the short spacers.
• Pre-drill when you are within 2" of the end of a board and going in from the side (e.g. all of the sub-assembly nails/screws and the 4 corners). Other than that, you can avoid pre-drilling, but it may help you keep a straight line if this is your first build.
• Metal brackets to hang the joists will give you a stronger connection. They should include directions for the type of fasteners, which will likely be shorter but very thick nails.
• Nails will have more shear strength, but screws will be less likely to pull out. If you have a sub-floor going on top of this, securing everything together, then I'd go with nails. I'd be lying if I didn't say I'd put in at least a few deck screws in each corner.
• 3 1/2" is a good length for any nails/screws. That gets roughly half of the fastener in each piece of lumber. When in doubt, get a longer/thicker fastener.
• Any untreated nails or screws will react with the chemicals in pressure treated lumber and quickly corrode. Hot dipped galvanized nails or treated deck screws should be ok, but check the label on the box to be sure it will work with the chemicals in your lumber.
• For each joint, a 2x4 would take 2 nails, a 2x6 would take 3, etc. Stay about 1" away from the edge of the board and space the nails about 1 1/2" - 2" apart.

It probably depends on how much effort you're going to be putting into it.

If you're just planning on slapping something together quickly, then sure, go for untreated, and if something goes wrong, you can rebuild it in a couple of years.

I'd personally use pressure treated, as I'd have to make a larger structure due to local ordinances. (I can't store firewood within 18" of earth; I don't know if it's an attempt to deal with an invasive species that caused a ban on exporting firewood from our county) Because of the extra work involved, any savings on cheaper materials would be wasted if I have to rebuild.

I'd also be interested in the anti-insect properties of pressure treated, not only the anti-rot aspects.