The mayor's office said the proposed school on the Near North Side would no longer be named after Obama.
Principal Maureen Ready said the cuts will force her to lay off three teachers and consolidate grades.
The grandmother of Mckenzie Phlipot, 12, said Tuesday she "does not agree" with CPS' conclusion.
Dyett High School supporters said they'd met with a sympathetic Mayor Rahm Emanuel, but nothing changed.
CPS performed well in Newsweek's new rankings of top high schools serving low-income students.
Schools on the Northwest Side are making up for budget cuts by relying on fees paid by parents.
Students will get a chance to make their case against the dress code at a town hall on Sept. 11.
Gov. Pat Quinn added $10 million in state funding to bring the number of workers to nearly 2,000.
The school also has considered expanding its building on Loyola Avenue to stay in the neighborhood.
School, on academic probation 11 years, has had two principal changes in less than a year.
The renovations to Taft were "long overdue," said Chicago Public Schools CEO Barbara Byrd Bennett.
Dyett High School was scheduled to close at the end of the upcoming school year.
The school hired seven teachers to keep up with continued growth.
Local 134 of the International Brotherhood of Electrical Workers agreed to hire student graduates.
Construction preperations on the annex could begin as early as Monday.
Get Our Daily Chicago Newsletter
Thank you for signing up!
Would you like to sign up for additional newsletters?
Thank you for signing up for additional newsletters.