Burnt and Frizzy Hair
A: The first thing you need to do is tell your daughter that if she doesn't do something to protect her hair from the damage she's doing to it, having greasy hair will be the least of her concerns. If she actually has a problem with oily hair, then she needs to address that issue and treat it, as opposed to overcompensating by drying the hair out.
Furthermore, when planning to heat style the hair (blow-drying or flat ironing) you should also use a leave-in conditioner to protect the hair from damage. Be sure to read our articles on thermal styling for additional information on the proper procedure for straightening the hair. The most important thing to remember is that the hair should be fully dry before using the flat iron.
In your daughter's situation, given that you state she dries her hair at night, she should blow-dry her hair to a "mostly dry" state before bed and allow it to finish drying overnight. In the morning, she can use the flat iron to smooth the hair. Doing this would allow her to have the hair completely dry as needed for flat ironing without over-drying the hair with the blow dryer.
As for color: if your daughter's hair is showing signs of damage, I would advise you to make her wait until her hair is in better condition before having any chemical service done.
I would also suggest that given her age and the fact that permanent hair color will require a commitment to maintaining the hair color or having additional color services to change the color, you insist that your daughter use semi-permanent or demi-permanent hair color in order to do minimal damage to the hair and allow her to simply let the color wash away when she grows tired of it.
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See also:
Flat iron styling
Heat styling basics
Damaged hair
How badly will straightening damage my hair?