I personally don't understand how you could put a mute under the strings and not effect the tuning/intonation. I haven't tried weather stripping, foam, or a sponge, though I have tried various sizes of socks and whatever I try sharpens the pitch. By putting anything underneath the strings and having them touch this material you would shorten the string length, so in theory wouldn't it sharpen the pitch no matter what you use?
I guess the key is to have mute low enough where the bridge is still the main point of contact but the foam just stops the string when it vibrates, but I would still think that would effect the tuning.
Sorry if I seem like I'm putting up an argument, I'm truly not, I would just love to put in a mute without affecting the tuning, but I haven't been able to find a way to do it yet. And I know it can be done because some of you say you've done it, I just don't quite understand the science behind it. Though I guess should explore other materials before I give up. I do believe that the density of the mute's material would vary the amount of sharpening to the pitch.