Does the relationship between plaintext and ciphertext moduli affect the security of BGV/BV SwHE?
This is necessary for security. Consider if $q$ was a multiple of $t$, so $q = v\cdot t$. Take your ciphertext $(a, \langle a, s \rangle + t\cdot e + m)$ and multiply through by $v$. You now have 
$$(v\cdot a, \langle v\cdot a, s \rangle + v\cdot t\cdot e + v \cdot m) = (v\cdot a, \ \langle v \cdot a, s \rangle + v\cdot m) \mod q$$ 
Suppose you're playing the CPA game, so there are only two possible values for $m$, which you know, so just try subtracting $v\cdot m$ from the second element and solve the "learning without errors" problem using your favorite algorithm to solve a linear system of equations (Gaussian elimination will work for BV &amp; BGV). 

For a more general answer, if $t$ is a zero-divisor in the ring $\mathbb{Z}_q$, the above attack will work, since you can kill the error by multiplying through by a non-zero value. By mandating that $t$ and $q$ be coprime, the only way to kill the error is to multiply by a multiple of $q$, which is 0 in the ring $\mathbb{Z}_q$.