Inside the batterydoor and the memorycarddoor there are little actuators for safetyswitches. If one of these Doors are opened, the camera Will NOT power up.
Perhaps one of the actuators is missing/ broken, or one of the switches inside the camera is defect.
Hi Susan. I'll take a stab at this because I had a similar problem once with my T5i. I removed my battery grip and inserted a charged battery into the camera. The camera would not power up. Turns out there is a limit switch (very tiny) that gets depressed when the battery door is closed. I hadn't reinstalled the battery door after removing the grip so the camera thought the battery door was open and would not power up. There's a similar switch on the card door too. It could be that one of those two switches has failed, or one of those doors is not closed. If both doors are in place, then one of the switches may have failed and the camera needs a repair. OR, the little tab on the battery door that depresses the limit switch has broken off. In that case you can stuff a tiny piece of paper into the hole to depress the switch and use the camera. You could order a new door in the mean time.