A B757-200 flight crew experienced TCAS false warnings at FL330; and shortly thereafter noticed a strong odor. They diverted to the nearest suitable airport.

Narrative

At cruise; FL330 when with no warning an aural TCAS 'Traffic' warning sounded followed shortly thereafter by a 'Descend' RA. The Captain and I simultaneously clicked off the autopilot and pushed the aircraft over. We descended to approximately FL326 before the RA went away. The Captain notified ATC of the RA. The target then disappeared completely from TCAS. We climbed back up and re-established cruise; autopilot on; at FL330. ATC advised no primary or secondary targets were nearby. We were discussing the possibility of a false warning or a military aircraft when the TCAS RA sounded again. This time we looked closely at the display; noticed the target was supposed to be slightly ahead and to the right at FL333. After a careful scan and based on the previous false alert we elected to turn the TCAS to TA only. A few moments later we both detected an acrid; electrical burning smell and immediately donned our masks and smoke goggles. The Captain pulled out the QRC and I took over ATC communications. After running the first steps of the checklist; the odor appeared to subside but returned quickly and stronger. Following the checklist we decided to land as soon as possible. The divert page showed ZZZ as the closest airport. I advised ATC while the Captain continued with the checklist. We were given direct ZZZ and a descent to FL320. Initial communication with the cabin indicated no fumes; however at around FL240 on the descent the purser called to indicate a strong odor in the cabin. The Captain ran the smoke/fumes elimination checklist which was successful. The Captain taxied clear and stopped to allow emergency response vehicles to inspect the aircraft for smoke or fire indications. No indications were present and the smell had subsided in the cabin so we taxied to the gate.