How high can a washing machine pump it's drain water out? We want to move our washing machine to the basement and wonder if we can get away without installing a sump pump.

How high will it have to pump? This won't be your biggest concern. Starting from the bottom of the discharge hose a check valve will have to be installed next to the exit point of the hose so when the pump stops the dirty water won't drain back into the washer. And that's just for openers. Next you will have to cut in a trap in the main and run a vent from the trap to the roof or revent back into a existing dry vent. Wait! You're not done yet. Because connecting into a overhead sewer won't leave you enough room for a 38 inch stand pipe out of the trap you will have to install a compression fitting on the trap raiser for the hose and make it a closed system.
Still want to move the washer to the basement? Regards, Tom