If you are going from the fretted note to an open note then it's just a matter of technique. Mute the string with your lifted finger. If you need to move the finger to another position like when changing a chord then perhaps you are accidentally doing a pull off where you are plucking the string with the fretting hand. Practice plunking the string and removing your finger without getting that plucked sound.
You give little information but if you are using an electric and running it thru some hi-gain distortion then you are going to have to learn various ways to mute unwanted string noise.
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