How much damage would a space probe cause?

Not enough to cause a global extinction.

I could plug things into equations, but I'm lazy. Plugging relativistic kinetic energy of a 100kg mass traveling at 0.4c gives $8 \times 10^{17}$ Joules. The page on the impact that is hypothesized to kill the dinosaurs, the Chicxulub impact, says that the energy released in this event was $4\times 10^{23}$ Joules.

A quick google search shows that the juno probe is about $3000$kg, or about thirty times heavier than the mass I plugged in. This gives that your probe would release 0.00006 times the amount of energy as in the Chicxulub impact. This is a lot, but not in the grand scheme of things!

Let's assume it travels at a constant speed of .4c (since we don't know anything about its acceleration) and its mass is 100kg. At those speeds, the relativistic effects are minor compared to the fact that I just totally made the mass up out of thin air, so we can do a classical solution.

Using $E=1/2mv^2$ we find that the energy is $1/2 \times 100 \times 119916983^2 = 7.19004143 \times 10^{17} ~\mathrm J$. Using my favorite page on the internet, Orders of Magnitude (Energy), we see that that's on par with detonating the Tsar Bomba, the largest nuclear weapon ever built. It's 6 orders of magnitude off from the energy released by the Chicxulub meteor event that is believed to have caused the extinction of the dinosaurs.

Thus there's little risk of an extinction event. That and hitting a planet by accident 4 light years away without trying is pretty hard!

Calculating the damage done due to an impact is an imprecise business, but we could use the KT extinction event (that finished off the dinosaurs) as a benchmark. This impact had an energy of around $4.2\times 10^{23}$ joules. The question is then how fast our probe would have to be going to deliver this much energy.

For a relativistic projectile the total energy is given by:

$$ E^2 = \frac{m^2v^2c^2}{1 - v^2/c^2} + m^2c^4 $$

and the kinetic energy is just this energy minus the rest mass energy $mc^2$.

Let's assume our probe has a mass of a ton ($10^3$ kg), which is around the mass of the Voyager probes. In that case to match the energy of the Chicxulub impact would need a speed of around $0.99999998c$. You're suggesting a speed of $0.4c$, and at that speed the kinetic energy would be around $8\times 10^{18}$ joules, which is a factor 50000 lower than the Chicxulub impact energy. Even so I wouldn't like to be standing underneath it when it hit.

But there are a couple of points that need to be made.

Firstly space is big - really big - and it's mostly empty. The chance of a probe hitting anything is so ridiculously small that no-one is going to take it seriously.

Secondly NASA does care what happens to its probes and this is part of the mission planning. For example the Cassini probe is going to be deliberately crashed into Saturn where the heat of reentry will destroy it (and any Earth life forms that might be hiding on it).