Can black acoustic sponge tiles be died to white?

Whatever you need to know can be found here, including how to make user friendly traps:
http://www.realtraps.com/articles.htm

I wouldn't paint, dye or try to bleach black acoustic foam white. You can just buy white acoustic foam tiles. Google search "white acoustic foam". There are several suppliers on the net that sell white acoustic foam tiles in many sizes and absorption patterns. Here is a link to one that I found.
http://www.silentsource.com/afoams-hushfoam.html

Usually the color of acoustic tiles is changed with paint or stain. If the paint is thick and gloppy it will cover up the absorbing holes and fissures and spoil some or all of the acoustic properties. Covering black with white would seem to need a thick coat of paint.

Other possibilities - covering the tile with fabric.

Then there is the high road - buy new!

Absorbing material needs to be stuff that is intended for the purpose.

This ain't.

The benefits of acoustic treatment is roughly based on the percentage of the room area covered and the quality of the absorbing material. You get extra points for putting stuff where it does the most good, but you are going to get benefits by the square foot - many of them.

If fiberglass and mineral wool give you problems, there is always cotton batt based products that are sold for the purpose:

http://www.audimutesoundproofing.com/acoustic-material.aspx

...and competitive products. Google is your friend!

Not so. By and large there's no functional difference between acoustical foam and open cell urethane mattress or packing foam. The linked foam will work just as good as acoustic foam. Memory foams typically are of much higher density, and therefore will work even better in the low end.
What is a consideration is fire resistance. Most acoustical foams are fire rated. Not all foams are, so when being used for a room treatment one should make sure the product is fire rated.