[Locked, wrong forum] clickable image now leaves blue lines

What's the URL to your online page? 

As Jon says, image maps are not responsive.  Used widely in the late 20th century when everyone had same sized computer displays, image maps are rarely used today. 

If you can work with code, an alternative approach is to build linked SVG shapes on top of a responsive raster image.  See working example below.   In smaller viewports, the map remains responsive.

https://jsfiddle.net/NancyO/6k3v4ge1/13/

Otherwise, I stay clear of image maps.  Separate linked images or buttons are a lot more flexible.

I agree with the others here, it's typically bad form to remove the outline from image maps, from an Accessibility point of view.

I'm guessing the reason you truly want it gone is because it doesn't line up with the image. When you use a standard image map on a responsive image and the image either grows or shrinks away from the size you used to create the map, the <area> tag's coordinates no longer line up with the image itself and cause the offset look you have posted above. The larger or smaller it gets, than the size used to set the coordinates in DW, the worse it gets. Even disappearing from the image entirely in certain cases

You should look into a responsive image map script to use javascript/jquery to take care of the drifing problem. When the outlines always line up with the image parts they're meant to cover, it's not nearly as distracting or noticeable to your viewers.

Something like this works very well: http://mattstow.com/experiment/responsive-image-maps/rwd-image-maps.html

That outline visually indicates the linked area, which is important for accessibility reasons (think of someone physically impaired who cannot use a mouse, but only 'tabs' through all the available links on a page, for example).

To remove it set the outline to 0 or none. But it considered VERY bad usability and accessibility practice to do so.