EEPROM 25LC1024 issue with PIC32MX OLIMEX board
It doesn't look like anything code-related. It looks like either a pin shorted to another output pin which is set to zero ( or another slave not tri-stating its MISO ). Disconnect the memory chip, set PIC MISO to output and try to toggle it by writing 1 and 0 to the corresponding port bit.

Another suspect is power supply to the memory chip. Make sure it receives VCC and has a capacitor from VCC to ground.