OK, let's get back to the original question: Is the image $T(S)$ of a linearly dependent set $S$ under a linear transformation $T$ linearly dependent?
We must to notice that, there are many possible definitions of span, linear combination and linearly dependence/independence, each of them again have several versions, one talking on $v_1,v_2,\cdots,v_k$(this vector can be required to be totally distinct, or not necessarily distinct, which means there are again two possible choices one may choose), one talking on a set, especially having the ability to talk on infinite set of vectors, such as $\{1,x,x^2,x^3,x^4,\cdots\}$.
First, suppose we adopt the version of span, linear combination and linearly dependence/independence that talking about $v_1,\cdots,v_k$(not set) which are not necessarily distinct. Then in this case, for example, we may say $v,v$ is linearly dependent. Then it is true that for any $v_1,\cdots,v_k\in V$ not necessarily distinct, if they are linearly dependent, then $T(v_1),\cdots,T(v_k)$(possible duplicated here) are linearly dependent.
However, the important point is that, the original proposition in the title, as one might not expect, is false. This is because the set language has an essence that repeating the same object in a set doesn't behave differently than the original set. I just thought and found an unbeatable counterexample.
Let $T:\Bbb R^3\to\Bbb R^3$ defined by $T(\mathbb{x})=\begin{bmatrix}3&0&0\\0&2&0\\0&0&0\end{bmatrix}\mathbb{x}$.
Let $S=\{(1,2,3),(1,2,1),(1,3,7),(1,3,5)\}$. Then since $S\subseteq \Bbb R^3$, then $S$ must be linearly dependent.
Next, by the definition of image of a set under a function, we see that $T(S)=\{(3,4,0),(3,6,0)\}$. Notice that this result followed by the definition of image of a set under a function, which has nothing to do with linear combination or span or linearly (in)dependence. So it doesn't depend on what kind of defintion you wish to choose! Then, as you can see, $T(S)$ is literally and undoubtfully linearly independent. So the original proposition in the title is FALSE.