Well, if it's burning coolant it'd be either a cracked head or a bad head gasket. Are you sure it's burning coolant, and not leaking it? Find out if there's oil in the coolant or coolant in the oil, like babymog said. So did it overheat after it started losing coolant, or start loosing coolant after it overheated?

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1987 300D, arctic white/palomino--314,000 miles
1978 240D 4-speed, Euro Delivery, light ivory/bamboo--370,000 miles
2005 Jeep Liberty CRD Limited, light khaki/slate--140,000 miles
2018 Chevy Cruze diesel, 6-speed manual, satin steel metallic/kalahari--19,000 miles
1982 Peugeot 505 diesel, 4-speed manual, blue/blue, 130,000 miles
1995 S320, black/parchment--34,000 miles (Dad's car)