No. Coolant flows through the core whenever the engine is turning over.
Replacing the core is not 'tough', and look on Ebay for a 'cheap' new core for around $50.
Leave the covers off both sides of the center stack and place shop towels or paper towels next to the leaking core to absorb the coolant. Getting coolant out of the carpet is a pain and if the coolant gets to the underside of the rubber mat and into the foam, almost impossible to dry out or clean up.
http://www.ebay.com/itm/Volvo-Heater-Core-850-S70-V70-C70-1994-through-2000-lifetime-warranty-/360771828487?pt=Motors_Car_Truck_Parts_Accessories&fits=Make%3AVolvo%7CModel%3A850&hash=item53ffad3b07&vxp=mtr#ht_8675wt_944
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