Can I run my washing machine drain line into a condensate pump so it drains better?
I think you have a couple of factors to consider in using a condensate pump. 

One is the volume of water per unit time relative to what the pump can handle. A condensate pump is going to expect a trickle flow from the air conditioning system. 

When used as a washing machine pump, the float will rise as the reservoir fills and activate the pump. Will the condensate pump remove water at a rate to match or exceed the washing machine?

Washing machines work in cycles. If the reservoir is too small to handle a maximum load cycle and the pump rate is too low, flooding may result.

On the other hand, if you can supplement the reservoir in some manner, the pump rate becomes less critical.

I don't think the water "quality" is going to be a problem. The output of a washing machine is grit/sand/dirt/fibers. The heavy stuff is going to collect in the reservoir and the suspended stuff is going to be pumped. I have a filter on our washing machine that catches all the washed-out material that used to enter my septic tank. I'm amazed at the grit/sand quantities, but also at the shed fibers that are captured. The low cost pumps typically use an impeller which is resistant to problems related to suspended material found in a washing machine output.