Will your weapon level restart when you prestige on mw3?

No, unfortunately.

When you prestige, you lose everything but your titles and emblems. Luckily with Modern Warfare 3, they have implemented a system that slightly takes the sting out of prestiging.

Everytime you prestige, you are given a prestige token. With this token you can use it to give yourself 2hrs of Double Experience points for either your weapons or levels, some titles and emblems at certain prestiges or you can unlock 1 piece of gear in the whole game. Whether it be any primary or secondary weapon, a perk, any lethal or tactical grenade or a deathstreak.

And the best thing is that whatever you unlock will always be unlocked for everytime you prestige after so you do not have to unlock it again, except for the double Weapon/Level XP tokens, which expire after those two hours and you have to use another token to get them again.