Component of one signal along the another
Generally speaking, about the only restriction that the signals need satisfy is that they have finite energy or power.  Since you mention vectors by which is presumably meant $N$-tuples of real or complex numbers and dot products, the restriction is the trivial one that each component of the $N$-tuple has finite value.

Practically, there are several issues that you need to think about depending 
on what you are trying to do.  Obviously
$$\langle \mathbf x, \mathbf y \rangle = \sum_{n=1}^N x[n](y[n])^*$$
is the dot product but how accurate a measure it is of the inner product
$$\langle x(t), y(t) \rangle = \int_{-\infty}^{\infty} x(t)(y(t))^* \mathrm dt$$ 
of the continuous-time signals $x(t)$ and $y(t)$ from which these samples 
came depends on whether the signals were filtered before sampling, the
sampling rate, whether the samples span the entire time interval where
the signals are nonzero, whether windowing was applied to the samples, etc.
You might be interested in the cross-correlation function $R_{x,y}(\cdot)$
instead of just the inner product (which is $R_{x,y}(0)$) in which case
FFTs might be used and you have to make sure that the signals are properly 
zero-padded so that you get the right answers. Finally, we began with
the "trivial" restriction that each component is finite which always holds
in practice, but if the components are large in magnitude, you need to worry
about  the possibility of overflow in the inner product computation.

So, as you can see,
in theory, theory and practice are the same, but in practice, they are 
different.