Should my water heater drip constantly?
Niall is correct.  Opening the T&amp;P valve to drain water from the water heater is a common practice, but if it's really old, it can be damaged by sediment and corrosion.  Where the plumber went wrong is not using the T&amp;P valve to drain some water, but rather trying to hammer it shut.  T&amp;P valves prevent BLEVEs (Boiling Liquid Expanding Vapor Explosion), which are a type of catastrophic explosion -- in this case, in the event of a malfunction in the water heater.   Let's just say that if your water heater BLEVEs, you can kiss your entire house goodbye, so since there is clearly a problem with your T&amp;P valve, you need to get yourself a new one.

In order to replace the T&amp;P valve, you can call a plumber or do the following:

1) Turn off the cold water ball valve.
2) TURN OFF THE WATER HEATER'S BREAKER IN YOUR BREAKER BOX (if the top heating element hits air, you can damage it, and it will make a massive shotgun-like sound)
3) Open some hot water outlets, such as the hot water handles on a few sinks.  These will allow air into the top of the hot water heater.
4) Run a garden hose from the hot water heater's drain to the street/driveway.
5) Open the water heater's drain, and empty the hot water heater below the T&amp;P valve.
6) If it does not flow on its own, you have sediment build-up (which you most likely do if your T&amp;P is failing.)  If it doesn't flow, open the cold water ball valve so that the water pressure can help break through the sediment layer.  After 2-3 minutes, close the cold water ball valve again to allow the top of the water heater to fill up with air.
7) Replace the T&amp;P valve.  Use teflon tape to help create a seal.

8) Turn off the drain.
9) Remove the garden hose, but keep it held high so no water comes out the end that is in your hand... and take it back outside.
10) Open the cold water ball valve.
11) A mixture of water/air should come out of the still-opened sinks after 1-2 minutes.  You can shut them off at this point if you don't mind seeing the air bubbles the next time you use the sink.
12) Turn the water heater's breaker back on.