The line of credit is a second deed of trust (mortgage). That means that it was recorded in second place behind your first mortgage. If you pay off the first mortgage with your refinance money, the credit line moves up to first-place position.
No lender will refinance your main home loan and accept second place. Accordingly, many credit line lenders will agree to file a document among the land records that states that they agree to accept second place behind the new lender. That document is called a subordination agreement.
There is no legal reason why any credit line lender must agree to subordinate. In fact, many such lenders have started to either restrict the amount of money that can be tapped from their loans or have canceled them outright. Why? Because as the equity in your home decreases, their security becomes threatened.
Have you or your refinance lender talked to that lender? Have they given you any valid reason why they do not want to subordinate?
Talk with your proposed new lender. Ask whether they make credit line loans. If so, you can refinance, get a new first mortgage as well as a new line of credit, and pay off the old lender.
Of course they would like their line paid off and closed out. In this current economic environment it only makes sense. It would be interesting to see some statistics on this rather than just the anecdotal stories we come across.