Regardless of some people thinking it might actually hurt your car in the long run, they are wrong. Only short term driving would be effected. You might notice blue smoke from your tail pipes from the burning of the oil in the mixure.
Now I don't know exactly how much you put in, but I would figure that the mixure of oil to gas in for your 'weedeater' is more than likely around 1:25. That is a very small amount of 2 cycle oil to be added (assuming you put a gallon or two into your tank) to a 10-15 gallon tank.
If you haven't already, fill your car completely with premium. You will not suffer any kind of long term effects as some may think. More than likely if it was only a small amount gas (a gallon or two), you probably won't even notice any kind of difference if you have a full tank of gas.
So in short, don't worry it will not damage your car's engine or fuel system nor will you need to pay a mechanic to "anything special". They would more than likely just take your money, and sift the gas out of the tank.