How close would the orbit of Mars have to be to that of the Earth to induce a collision? When the Earth passes by Mars they exert a slight gravitational pull on each other, but the force is so small that their orbits are not significantly disturbed.

However, if we imagined that the orbits of the Earth and Mars were coplanar and much closer to one another, then there would presumably be some critical distance at which they would move together and collide.

What is that critical distance? In other words, if we shrunk the radius of the orbit of Mars, how much would it have to shrink before a collision would occur by the natural force of gravity between the two bodies?
It's more likely that perturbations by another planet might induce a collision.

In other systems with bodies in almost the same orbit, what happens is like a game of orbital catch-up: one overtakes the other, then they swap places, so in your (rather vague) scenario, that might well be the outcome instead of an outright collision.

Some of the Saturn ring moonlets interchange positions, for example.

The solar system is stable over a timeline of millions of years (as far as we know), but there have been instabilities in the past, and due to "butterfly effects" instabilities could occur in the future.
(See also https://plus.maths.org/content/rocks-suggest-theres-chaos-solar-system)

Have a play with a simulator such as orbitsimulator.com !

http://orbitsimulator.com/gravitySimulatorCloud/simulations/1433364162018_Solar%20System.html