I have a washing machine I want to relocate to the other side of the laundry room. There is a sink that's coming out where the washer is going in. The drain line is 1 1/2". Will this be a problem.

Yes, it will be a problem. Today's washers discharge large amounts of water in limited time frame. Overflow occurs. Pipe should be at least 2". It can be vented with 1 1/2" pipe. But that said: in older houses we see 1 1/2" drains for washers - and it works. So call is yours. If you never had any problem with drainage with that sink, you can risk it. If you had - than I would strongly suggest to upgrade the drain to 2".