Tennenbaum's theorem says neither addition nor multiplication can be recursive in a non-standard model of arithmetic. I assume recursive means computable and computable means computable by a Turing machine (TM).
The 3-state TM described below takes the unary representation of two natural numbers as input and outputs the unary representation of the sum of the two numbers. A unary representation is a string of 1's (possibly empty) followed by a 0. Given the input 110110 this TM will output 111100.
This TM halts on any input tape with two 0's. It can add any two standard natural numbers in a finite number of steps. It must also work inside any non-standard model. Assume it didn't. Then we could define the set of numbers, $x$, such that this TM correctly adds $x+x$. Assuming this set has no largest element, it would be an inductive set since it includes all standard natural numbers. A non-standard model can't have an inductive proper subset proving this TM must work in any non-standard model.
Another scenario is when we have a non-standard model in a meta-theory like ZFC. We must assume our meta-theory uses a standard model of arithmetic. The TM above still adds standard natural numbers, but Tennenbaum's theorem says there can't be an algorithm in the meta-theory that computes addition in this non-standard model.
Why can't the TM I describe above compute addition in this non-standard model? Is it simply because the input tape would have to be infinitely long if one of the input numbers is an "infinite" number?