Perhaps the first, but not the second comma.
I suggest 'His ID card is missing, as well as his dog Ian; though it is likely no-one will notice the latter'.
Note I have replaced the comma after Ian with a semi colon. This is not essential, but there has to be some speech mark there in order to separate Ian from the second part of the sentence, which would, if you wanted it to, stand as a separate sentence altogether.
I tend to omit commas in simple sentences, where the message is obvious, such as 'Tiddles the cat died'.
With subordinate clauses you need a pair of commas, or in my example a comma and a semi-colon. These two commas partition the clause from the rest of the sentence. When pondering where to put the commas remember that the sentence still has to make grammatical sense if the part between commas is omitted.