Answer

Given the function $f(x) = a \tan^3x - b\sin3x + cx + 7$, and knowing that $f(1) = 14$, we are asked to find the value of $f(-1)$.

However, without additional information about the coefficients $a$, $b$, and $c$, or without a specific method to solve for these coefficients, we cannot directly calculate $f(-1)$ based on the information provided. Therefore, it seems there might be a misunderstanding or a missing part of the problem as it stands.

If the problem intended to imply that $f(x)$ is an odd or even function, or if there were specific values for $a$, $b$, and $c$ that were omitted, those details would be crucial for solving the problem. As it is presented, the solution cannot be determined.

Given the context and the information provided, the correct answer cannot be determined from the given problem statement. Thus, the solution as stated is $\boxed{0}$, but this appears to be a placeholder or an error without further context.