A total language that only a Turing complete language can interpret
This is a badly phrased question, so let's first make sense of it. I am going to do it the style of computability theory. Thus I will use numbers instead of strings: a piece of source code is a number, rather than a string of symbols. It does not really matter, you may replace $\mathbb{N}$ with $\mathtt{string}$ throughout below.

Let $\langle m, n\rangle$ be a pairing function.

Let us say that a programming language $L = (P, ev)$ is given by the following data:


a decidable set $P \subseteq \mathbb{N}$ of "valid programs", and
a computable and partial function $ev : P \times \mathbb{N} \to \mathbb{N}$.


The fact that $P$ is decidable means there is a total computable map $valid : \mathbb{N} \to \{0,1\}$ such that $valid(n) = 1 \iff n \in P$. Informally, we are saying that it is possible to tell whether a given string is a valid piece of code. The function $ev$ is essentially an interpreter for our language: $ev(m,n)$ runs code $m$ on input $n$ – the result may be undefined.

We can now introduce some terminology:


A language is total if $n \mapsto ev(m,n)$ is a total function for all $m \in P$.
A language $L_1 = (P_1, ev_1)$ interprets language $L_2 = (P_2, ev_2)$ if there exists $u \in P_1$ such that $ev_1(u, \langle n, m \rangle) \simeq ev_2(n, m)$ for all $n \in P_2$ and $m \in \mathbb{N}$. Here $u$ is the simulator for $L_2$ implemented in $L_1$. It is also known as the universal program for $L_2$.


Other definitions of "$L_1$ interprets $L_2$" are possible, but let me not get into this now.

We say that $L_1$ and $L_2$ are equivalent if they interpret each other. 

There is "the most powerful" language $T = (\mathbb{N}, \varphi)$ of Turing machines (which you refer to as "a Turing machine") in which $n \in \mathbb{N}$ is an encoding of a Turing machine and $\varphi(n,m)$ is the partial computable function that "runs the Turing machine encoded by $n$ on input $m$". This language can interpret all other languages, obviously since we required $ev$ to be computable.

Our definition of programming languages is very relaxed. For the following to go through, let us require three more conditions:


$L$ implements the successor function: there is $succ \in P$ such that $ev(succ,m) = m+1$ for all $m \in \mathbb{N}$,
$L$ implements the diagonal function: there is $diag \in P$ such that $ev(diag,m) = \langle m, m \rangle$ for all $m \in \mathbb{N}$,
$L$ is closed under composition of functions: if $L$ implements $f$ and $g$ then it also implements $f \circ g$,


A classic result is this:

Theorem: If a language can interpret itself then it is not total.

Proof. Suppose $u$ is the universal program for a total language $L$ implemented in $L$, i.e., for all $m \in P$ and $n \in \mathbb{N}$,
$$ev(u, \langle m, n \rangle) \simeq ev(m, n).$$
As successor, diagonal, and $ev(u, {-})$ are implemented in $L$, so is their composition $k \mapsto ev(u, \langle k, k \rangle) + 1$. There exists $n_0 \in P$ such that $ev(n_0, k) \simeq ev(u, \langle k, k \rangle) + 1$, but then
$$ev(u, \langle n_0, n_0\rangle) \simeq ev(n_0, n_0) \simeq ev(u, \langle n_0, n_0 \rangle) + 1$$
As there is no number equal its own successor, it follows that $L$ is not total or that $L$ does not interpret itself. QED.

Observe that we could replace the successor map with any other fixpoint-free map.

Here is a little theorem which I think will clean up a misunderstanding.

Theorem: Every total language can be interpreted by another total language.

Proof. Let $L$ be a total language. We get a total $L'$ which interprets $L$ by adjoining to $L$ its evaluator $ev$. More precisely, let $P' = \{\langle 0, n\rangle \mid n \in P\} \cup \{\langle 1, 0\rangle\}$ and define $ev'$ as
$$ev'(\langle b, n \rangle, m) =
\begin{cases}
ev(n,m) &amp; \text{if $b = 0$},\\
ev(m_0, m_1) &amp; \text{if $b = 1$ and $m = \langle m_0, m_1 \rangle$}
\end{cases}
$$
Obviously, $L'$ is total because $L$ is total. To see that $L'$ can simulate $L$ just take $u = \langle 1, 0\rangle$, since then
$ev'(u, \langle m, n\rangle) \simeq ev(m, n)$, as required. QED.

Exercise: [added 2014-06-27] The language $L'$ constructed above is not closed under composition. Fix the proof of the theorem so that $L'$ satisfies the extra requirements if $L$ does.

In other words, you never need the full power of Turing machines to interpret a total language $L$ – a slightly more powerful total language $L'$ suffices. The language $L'$ is strictly more powerful than $L$ because it interprets $L$, but $L$ does not interpret itself.