If someone bored right through the Earth’s crust to the mantle, would it cause a volcanic eruption or what would result?

Probably not.

In most cases, your hole would collapse without reinforcement before you hit the mantle. Or you dig a wide enough hole (think about as wide as it would be deep) to prevent collapse.

Next factor: pressure. To get an eruption you need to find a pocket of high pressure. Now comes the tricky part. How to release the pressure at, or nearly at, the same time. Most likely you’d be progressively releasing the pressure as you dug towards the pocket.

I think the best you could hope for is an oil derrick type gusher. Not really an eruption. Or an explosive base penetrator that can dig fast enough to do a “pop the balloon” scenario. Either one would require a great deal of study of the ideal area and engineering to deploy the method of penetration.