Many things can cause a spike in your insurance premiums. If you get into an accident and you're at fault, you can expect to pay more. If you lend your car out and your car is involved in an accident, you could end up paying more. If you're caught speeding, you can expect to pay more -- especially if it happens multiple times. Most insurance companies don't deal with a speeding ticket quite the same if someone else is driving your car, however, as long as the driver is not a named insured on the policy and doesn't drive your car regularly.
It's On Him...Probably
If you let someone borrow your car and they are caught speeding, your insurance premiums will not generally be affected. The driver will have to pay the ticket and will probably have to pay more for his insurance. In most cases, you will not be affected negatively. However, insurance policies vary considerably. While most insurance companies will not change your rates due to someone else speeding in your car, you should check the terms of your policy or call your agent if someone is caught speeding in your car, just to be sure.