Yes you do need another "to". Let me explain why.
You are making a comparison between two nouns: "NOUN1 is easier than NOUN2". "Put it together" is not a noun. That is why your sentence is incorrect.
To make your sentence work, you need to transform the idea of "put it together" into a noun. "Put" is a verb, not a noun. There are two magic ways that work to change any verb into a noun: (1) the infinitive, and (2) the present participle. "To put" and "putting" are both nouns. (They are also other things too, such as adjectives, but that doesn't concern us here.). Hence there are two ways to fix your sentence:
(1) "To take apart a computer is much easier than to put it together"
(2) "To take apart a computer is much easier than putting it together"