I have a 50 gallon natural gas HW heater that is original to my house (14 years old or so). I moved in six months ago and there is some dripping from the overflow pressure valve/tank, as well as some very minimal dripping from the valve handle that shuts off the water flowing out from the tank.
I have no idea if this tank has been properly maintained, including whether the anodes have been replaced.
Is it worth while to have a plumber look at the tank/overflow to see if I can get along with just a repair, or should I bite the bullet and replace it since I'll have to do it sooner or later?
If I replace it, it will require about $1000 -- new tank, removal of old, new overflow tank, city permits ($130).
Neither of the things you mention has any bearing on the condition of the water heater, unless the drip from the shutoff valve has been dripping onto the water heater for a long time. The drip from the temperature/pressure relief valve could either mean a faulty T&P or a thermal expansion issue. Read more about that farther down in The Tank at the subject Temp/Relief Valve. If it's just a bad valve, that's not a big deal.
I'd suggest you go to Know-how and follow the instructions for checking the combustion chamber or electric ports for signs of corrosion and damage and if all that looks good, consider checking the anode, if you can. Sometimes they're hard to get out.
And if you don't like what you see, go to Choosing a water heater so you can make an informed choice on your next heater.