They only heat up when in a large pile. Spread out now, or anytime, is OK.
I put down a light top dressing of chips anytime if the ground is damp.
I don't spread chips in the summer and early fall in areas where I don't water. It takes a lot of rain to penetrate a good layer of chips and get to the roots.
I also don't spread chips when the pile is 'smoking' with fungus spore. Usually that happens in about a week after dumping and can last a month. My lungs can't take it anymore.
Water vapor escaping from a pile doesn't bother me.
Mike

Can you explain the "smoking"? The 2 week old pile is hot and looks like it has ash in it. Is this fungus spore? I worried about the air borne particles too.

Water vapor is sometimes called steam by mistake. It's white.
Mushroom spore is darker and dry.
If your pile is two weeks old with a lot of leaves, it's probably mushroom spore and water vapor. :-(
Does it feel hot when you dig in to it?
Do you see mold?
Mike

Went out and checked the 2 week old pile and it is hot, 110 degrees and has water vapour. No sign of green leaves now. I'm going to spread the chips around my raspberries and on paths in the veggie garden. The other newer pile is going on my perennial bed. I love chips.
Just hope the weather stays dry for another day.

As mentioned the main problem is the lung hazard posed by mulch piles that are still generating clouds of mold. Wait until this dies down or wear breathing protection.

I've never had any problem with new chips burning anything. They don't "tie up" nutrients either, in fact as the material starts to break down it produces a sort of nutrient beer - I once stuck a yellowish hybrid rhododendron into a newish pile and the shrub greened up nicely.