Removing lines from a vector trace of a signature

Thanks for the sample file.

You may get a pretty clean result by using the Shape Builder tool and Shift dragging across the signature, followed by Object menu > Path > Simplify.

But I guess you just used inappropriate settings while auto-tracing the raster image. With appropriate settings it should create much cleaner results right from the start.

For example, this is what I get when auto-tracing (raster image, 300 ppi) with the default Black and White setting, expanding the live trace object and simplifying the paths a bit:

By the way, if auto-tracing your template raster file creates a mess of countless unnecessary and useless fragments in all cases, though you may think you are using ideal trace settings and probably have tried various pathfinder actions to unite things without success, it often just helps to apply one unique fill colour to the fragments (black is a good choice), then re-rasterize the trace and finally auto-trace the re-rasterized object again.