Should I say the project's status or the project status?
Project's status would be preferable in speech and formal writing, meaning the status of this particular project.
Ex: What's the project's status at this point?
If you were preparing a form or spreadsheet and the status of the project were a data point, project status would be an appropriate heading for the entry.
Project Name: Take Over World
Project Start Date: Yesterday
Project Status: In progress
"The project's status" is correct, as it is the status of the project you are talking about.
"Project Status" is also used, but normally as a headline or bullet point as it is not grammatically correct in a sentence.