Is it possible to dye mesh or tule (like a ballet tutu) ?

It depends on what fiber it is made of. A lot of tulles are made of easily-dyed nylon, but some are made of practically undyeable polyester.

If it is 100% nylon, then it will be easy to dye, using a type of dye called acid dye. You will need to heat the tulle in the dyebath, preferably in a cooking pot on top of the stove, though there are also instruction for dyeing nylon in the washing machine. The nylon requires a little bit of acid, such as ordinary white vinegar, in order to take up the dye well. Use 100 ml (about two-fifths of a cup) per gallon of water that you've mixed the dye into; for a 20-gallon washing machine load, if you're dyeing in the washing machine, use 8 cups of vinegar.

You can use an all-purpose dye, such as Rit, because it contains acid dyes. You can even use good tie-dyeing dye, such as Procion MX dye, if you substitute vinegar for the soda ash presoak. The most wash-proof dyes for nylon are the Lanaset dyes, but you can buy those only online, not in most local stores.

If, on the other hand, your tulle is made of a different synthetic fiber, such as polyester or acetate, you will not be able to dye it so easily. Polyester and acetate are fibers that cannot be dyed with ordinary dyes, only with a special kind of dye called disperse dye, and they require extensive boiling with the dye.