Linear non-causal system
If the input is zero up until t=to, must the output be zero before to because of linearity?

The output can be any function of the form
$$y(t) = \int_{-\infty}^{\infty}h(t-\tau)x(\tau)d\tau$$
and still be a linear (but not causal) system.
For a causal system, we'd have \$h(t') = 0\$ for \$t'&lt;0\$, and then if the input were 0 from \$-\infty\$ up to some time \$t_0\$, the output would also be 0 for \$t&lt;t_0\$.