No, you cannot normally apply any templates to a familiar.
A familiar is an animal chosen by a spellcaster to aid him in his study of magic. It retains the appearance, Hit Dice, base attack bonus, base save bonuses, skills, and feats of the normal animal it once was, but is now a magical beast for the purpose of effects that depend on its type. Only a normal, unmodified animal may become a familiar. An animal companion cannot also function as a familiar. (PRD, Wizard: Familiars)
The exception to this rule is contained within the Improved Familiar feat, which allows you to apply certain templates to your existing familiar based on your alignment and arcane spellcaster level. As written, Improved Familiar does not allow you to apply the Mutant template, however, it would be reasonable for a GM to allow it, since the Mutant template is fairly new and Improved Familiar was written before the template existed.
I would probably require Arcane Spellcaster Level 3 to match the other templates that can be applied to a familiar with the Improved Familiar feat.