The following links take you to a page with information on things to do in York for free (the list is in alphabetical order):
York Minster - unfortunately you need to pay to visit this magnificant church. But it is still a working church - and there are services at York Minster every day. You can see much of the church for free by simply going to one of the daily services. At the end of the service you can even wonder around for 5 - 10 minutes too.