Eventually, you'll get to the point where you're subconsciously muting strings all the time as you play. Over time, you start to develop little tricks and techniques like palm muting or muting individual strings with your right and left hands. So yes, it's a good idea to mute strings as you play...however, keep in mind that it might take a long time to remember all the various muting techniques for each lead you play.

What I usually do is play through a solo slowly and figure out the most efficient way to get around the fretboard. Then what strings make the lead sound more interesting (a G string sounds completely different than a B string playing the same note), then what noises are being made and how I can get rid of them.
Then I'll go through the lead enough times that I don't have to consciously think about techniques as I'm playing. You want to get to the point where you're able to get into the feel of what you're doing without the technical details getting in the way.