I have a`steady drip from my pressure relief valve on my Bryant natural gas heater

sounds like pressure relief valve is stuck closed free of or replace

It is normal for water to drip out sometimes from the pressure relief valve outlet pipe. If it drips continuously then open and shut the pressure relief valve several times so the pressure relief valve shuts off properly.
If this does not fix the dripping problem then the pressure relief valve is faulty and needs to be replaced.

1) If water flow stops when tap is off, then Sound of water running: could be factory-installed heat trap inserts or nipples: These things cause noise.
http://waterheatertimer.org/Low-hot-water-pressure.html#trap
http://waterheatertimer.org/9-ways-to-save-with-water-heater.html#heat-trap
http://waterheatertimer.org/How-to-troubleshoot-gas-water-heater.html

2) If sound of water continues when tap is off, then there is a broken pipe or leak somewhere in plumbing system.

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The exspansion tank is more than likly the problem. The temperature/presure valve may also be the problem. Replace both. To replace the exspansion tank turn off the supply water coming into the water heater. Open the relief valve and purge the presure off the tank. drain the water heater slightly below the temperarture/presure valve to replace. The exspansion tanks screws off. Use new teflon taope on the threads when replacing both the exspansion tank and the T&P valve. This work can be compleated by the homwowner with some mecanical inclination.

You'll obviously have to determine where leak is coming from, then we can decide how to stop it. If it's the pressure relief valve outside, and it's only a small drip, that is normal, but you can try lifting arm on relief valve and cleaning it out a little to see if that helps clean it out. BE CAREFUL when water is hot, that you don't burn yourself, thus better to try it when cooled down. If it's dripping, or running out too much, it may be that water is too hot, or water pressure is too high as well, so check that as well. Usually smart to install a pressure regulator in line from hose to trailer. If your running off on board pump, not very likely that pressure is too high as they are usually set at 40-50 psi, which shouldn't cause that. If water doesn't feel hotter than before, then best to replace pressure relief valve, if that's where dripping (excessive) is coming from.

It's gas operation only, but has electronic ignition for propane. ( just to confirm, if you look at back of water heater, it would have a metal or black plastic cover and hydro cable going in side or top of it.)
There are usually 3 things that may cause pressure relief to leak. 1) Faulty,corroded pressure relief valve, which usually needs to be replaced if it goes faulty. 2) water temperature is getting too hot, which could indicate a faulty thermostat. 3) If your hooked up to city water, the pressure may be too high, causing excess pressure build up in hot water tank, thus pressure relief opens. If your running off your on board pump, this is probably not the cause, as they are usually set at 40 psi.
Ist, - try pulling lever on relief valve slightly open ( careful it's hot !!) to clean out any calcium that may be logged on seal thus causing leak. If that doesn't do it, and the hot water isn't running out steady and appear to be under extreme pressure while leaking, then I'd change the pressure relief valve. Remember when you do, shut off water to system, and relieve pressure by opening a tap, and pressure relief as well after that. New one shouldn't be too expensive, just make sure you get same thread, 1/2"NPT or 3/4"NPT. Use a little teflon tape, or pipe dope whatever you prefer when re-installing as well as clean threads a little before.
If it really starts gushing out pressure relief when heating or heated, and water seems hotter than normal, it's more than likely a faulty thermostat. I'll wait for your reply on what condition your experiencing before I go in to how to change the thermostat.

Yes, Turn the knob on the gas valve to off. Turn off the incoming water line into the water heater. If you have a valve on the outlet turn it off too. Open the drain at the bottom of the heater to drain a little water from heater. You can then lift the lever on the temperature/pressusre relief valve to let a little air in to release the vacuum and let the tank drain. If the relief is in the side let it drain a bit longer so the water level is below the outlet. Be sure to re-install the down pipe on the relief valve on the new valve. You should drain the a few gallons of water at least once a year from your heater, and you should exercise your temperature relief valve once a year to be sure it will function. If your new valve leaks suspect your temperature control allowing the water to get too hot.

Yes it can be replaced. This is a temperature/pressure relief valve and is the second most important safety device on your heater. They rarely make racket. Is it leaking? if so yes replace it. Water heaters should have a portion of water drained annually along with operating the relief valve. IF you raise the handle a few times, and it continues to leak, then it should be replaced. If you replace it be sure to replace it with EXACTLY the same temperature and pressure rating that is on it now. Allow the water heater to cool, and drain pressure before removing. If it is rumbling inside, you might have calcium deposits bouncing around in the bottom. Drain some water and see how it looks. (Often if you operate drain, and haven't in a long time it won't reseal)

look can look to wear the water nis comeing from