I store a lot of my own tulips over summer and I wouldn't personally feel comfortable burying them in fine wood chips in a damp basement. It sounds to me that the wood chips would impede airflow and I have a hard time seeing how a few wood chips would be able to absorb all the excessive moisture from the whole basement area. I occasionally store bulb that are susceptible to drying out in dry sand and this is the normal reason to why you would cover bulbs with something during storage. That said the bulbs might still be fine if you just need to store them for a month.
In my own experience two things can happen to tulip bulbs in storage. If it is too warm and dry for a long time the bud inside the bulb might dry out on very sensitive cultivars (all tulips are not the same!), but the bulb itself will grow just fine, it just wont bloom. Moisture on the other hand will make them start to show signs of growth early, which is not a disaster, but prevents long storage.
Moisture and high temperatures in combination would probably be the worst, but this isn't conditions I tried for my own bulbs.
The thing to remember here is that the tulip bulb is usually actually happier above ground than in it during the summer months. However if your soil is really dry and is likely to stay so for a while, then nothing prevents you from planting the bulbs early.
If uncertain I would divide the tulip bulbs and try storing them in a few different areas. Spread the risks and gain some knowledge for the future.