I'm fairly confident that the following assertion is true (but I will confess that I did not verify the octahedral axiom yet):
Let $T$ be a triangulated category and $C$ any category (let's say small to avoid alarming my set theorist friends). Then, the category of functors $C \to T$ inherits a natural triangulated structure from T.
By "natural" and "inherits" I mean that the shift map $[1]$ on our functor category sends each $F:C \to T$ to the functor $F[1]$ satisfying $F[1](c) = F(c)[1]$ on each object $c$ of $C$; and similarly, distinguished triangles of functors
$$F \to G \to H \to F[1]$$
are precisely the ones for which over each object $c$ of $C$ we have a distinguished triangle in $T$ of the form $$F(c) \to G(c) \to H(c) \to F[1](c).$$
The main question is whether this has been written up in some standard book or paper (I couldn't find it in Gelfand-Manin for instance). Perhaps it is considered too obvious and relegated to an elementary exercise. Mostly, I am interested in inheriting t-structures and hearts from $T$ to functor categories $C \to T$, and would appreciate any available reference which deals with such matters.
Best Answer
The statement is false.
For example, take $C=[1]\times [1]$ to be a square and $\mathcal{T} = h\mathsf{Sp}$ to be the homotopy category of spectra. Now consider the square $X$ with $X(0,0) = S^2$, $X(1,0) = S^1$, and the other values zero, and the other square $Y$ with $Y(1,0) = S^1$ and $Y(1,1) = S^0$. Take the maps $S^2 \to S^1$ and $S^1 \to S^0$ to be $\eta$, and consider the natural transformation $X \to Y$ which is given by multiplication by 2 on $X(1,0)=S^1 \to S^1 = Y(1,0)$.
If this map had a cofiber, then, from the initial to final vertex we would get a map $S^3 \to S^0$. Following the square one direction, we see that we would have some representative for the Toda bracket $\langle \eta, 2, \eta\rangle$. Following the other direction, we factor through zero. But this Toda bracket consists of the classes $2\nu$ and $-2\nu$; in particular, it does not contain zero.
[Of course, this example can be generalized to any nontrivial Toda bracket/Massey product in any triangulated category you're more familiar with.]
Indeed, the Toda bracket is exactly the obstruction to 'filling in the cube' for the natural transformation $X \to Y$.
Anyway- this is one of many reasons to drop triangulated categories in favor of one of the many modern alternatives (e.g. stable $\infty$-categories, derivators, etc.).
As for t-structures and so on, in the land of stable $\infty$-categories these are easy to come by. (See, e.g., Higher Algebra section 1.2.1 and Proposition 1.4.4.11 for various tricks for building these.)