What is the minimum size of a ramjet?
Ramjets could be miniaturized, but at some point, air viscosity and boundary layer thickness will make it impossible for them to work. Simply put, the intake hole will get so small that air will find it easier to flow around the whole thing and ignore all of its complicated inner structure. 

In more technical terms, as the intake shrinks, intake velocity will approach an asymptotic limit regardless of flight velocity due to boundary layer friction, and for very small intakes, the air mass flow will be insufficient to sustain fuel combustion. That point seems to be somewhere in the centimeter range, but an engine becomes increasingly less useful as it approaches this point. Manufacturing parts this size isn't a problem even now.

Ramjet shells can help long-range artillery, but ramjet tank shells would not be effective in combat.

The figure of merit for tank rounds is their high-density penetrator's sectional energy at the target. Discarded sabot rounds are more effective at delivering this than any kind of engine internal to the projectile. Combat ranges are short enough to preserve the velocity.

If you had to use some sort of engine for a tank projectile, it would be a rocket. The reason is easiest to explain in numbers. At a range of 7.5 km with average 1.5 km/s velocity and a 10 kg round, to get 500 m/s of delta-V, you need 10 g = 1 kN of thrust (engine), plus 50 seconds of impulse (fuel). We'll ignore engine mass for the first approximation.

A rocket with TWR=50 and ISP=250 seconds will weigh 100 kg / 50 = 2 kg for the engine + 10 kg * 50/250 = 2 kg for the fuel, for 4 kg total.
A ramjet with TWR=5 and ISP=1000 seconds will weigh 100/5 = 20 kg for the engine + 10 kg * 50/1000 = 0.5 kg for the fuel, for 20.5 kg total.  

Simply put, when flight time is this short, a heavy engine like a ramjet doesn't pay for the minute amount of fuel (or rather oxidizer) mass it saves vs a light and powerful rocket.

This applies to all combat applications within visual range: rockets are better, and guns are better still if you can afford the launcher weight. Beyond visual range, rockets or hybrid gun/rockets take over, and past 100 miles, air-breathing engines become viable.